"14. Circles and Circle theorems "

# "14. Circles and Circle theorems " ## Circle vocabulary and properties ### Introduction Welcome to our first lesson on Circles and Circle Theorems! Circles are fundamental shapes in mathematics and appear everywhere in the world around us, from the wheels of a car to the orbits of planets. To understand circles deeply, we first need to learn the language of circles – their essential parts and properties. This lesson will introduce you to the basic vocabulary and characteristics of circles, which will form the foundation for more advanced topics. ### Key Concepts #### What is a Circle? A **circle** is defined as the set of all points in a plane that are equidistant from a fixed point called the **center**. #### Basic Parts of a Circle Understanding these basic components is crucial for describing and working with circles: * **Center:** The fixed point from which all points on the circle are equidistant. It is usually denoted by a capital letter, e.g., O or C. * **Radius (plural: radii):** A line segment connecting the center of the circle to any point on the circle. All radii of a given circle are equal in length. We often use the letter \( r \) to denote the radius. * **Diameter:** A line segment that passes through the center of the circle and has its endpoints on the circle. The diameter is twice the length of the radius. \[ d = 2r \] * **Chord:** A line segment connecting any two points on the circle. The diameter is the longest chord of a circle. * **Arc:** A continuous part of the circumference of a circle. Arcs are typically named by their endpoints and sometimes an intermediate point (e.g., arc AB, arc ACB). * **Minor Arc:** An arc shorter than a semicircle. * **Major Arc:** An arc longer than a semicircle. * **Sector:** A region of a circle bounded by two radii and the arc between their endpoints. * **Minor Sector:** The smaller region formed by two radii. * **Major Sector:** The larger region formed by two radii. * **Segment:** A region of a circle bounded by a chord and an arc. * **Minor Segment:** The smaller region formed by a chord and its corresponding minor arc. * **Major Segment:** The larger region formed by a chord and its corresponding major arc. * **Tangent:** A line that touches the circle at exactly one point. This point is called the **point of tangency**. * **Secant:** A line that intersects the circle at two distinct points. #### Circumference and Area * **Circumference (C):** The distance around the circle. It is analogous to the perimeter of other shapes. The formula for circumference is: \[ C = 2\pi r \quad \text{or} \quad C = \pi d \] where \( \pi \) (pi) is a mathematical constant approximately equal to \( 3.142 \) or \( \frac{22}{7} \). * **Area (A):** The amount of surface enclosed by the circle. The formula for the area of a circle is: \[ A = \pi r^2 \] #### Basic Properties of Circles * **Radii Equality:** All radii of the same circle are equal in length. * **Diameter as Longest Chord:** The diameter is the longest possible chord in a circle. * **Perpendicular from Center to Chord:** A line drawn from the center of a circle perpendicular to a chord bisects the chord (divides it into two equal parts). * **Tangent and Radius:** A tangent to a circle is always perpendicular to the radius at the point of tangency. ### Vocabulary * **Circle:** A round plane figure whose boundary (the circumference) consists of points equidistant from a fixed center. * **Center:** The central point from which all points on the circumference are equidistant. * **Radius:** A line segment from the center to any point on the circle. * **Diameter:** A line segment passing through the center with both endpoints on the circle. * **Chord:** A line segment connecting two points on the circle. * **Arc:** A portion of the circumference of a circle. * **Sector:** A region of a circle bounded by two radii and an arc. * **Segment:** A region of a circle bounded by a chord and an arc. * **Tangent:** A line that touches a circle at only one point. * **Secant:** A line that intersects a circle at two points. * **Circumference:** The perimeter or distance around a circle. * **Area:** The measure of the surface enclosed by a circle. * **Pi (\( \pi \)):** A mathematical constant, the ratio of a circle's circumference to its diameter, approximately 3.14159. * **Semicircle:** Half of a circle, formed by a diameter and the arc it subtends. * **Quadrant:** One quarter of a circle, formed by two perpendicular radii and the arc they subtend. ### Examples 1. **Identifying Parts:** In a given circle with center O, identify the following: * If OB is a radius, then any line segment from O to a point on the circle is also a radius. * If AB passes through O and its endpoints are on the circle, then AB is a diameter. * If PQ connects two points on the circle but doesn't pass through O, then PQ is a chord. 2. **Radius and Diameter:** * If the radius of a circle is 7 cm, its diameter is \( d = 2r = 2 \times 7 = 14 \) cm. * If the diameter of a circle is 20 cm, its radius is \( r = d/2 = 20/2 = 10 \) cm. 3. **Circumference Calculation:** * Find the circumference of a circle with a radius of 14 cm. (Use \( \pi = \frac{22}{7} \)) \( C = 2\pi r = 2 \times \frac{22}{7} \times 14 = 2 \times 22 \times 2 = 88 \) cm. 4. **Area Calculation:** * Find the area of a circle with a diameter of 10 m. (Use \( \pi = 3.14 \)) First, find the radius: \( r = d/2 = 10/2 = 5 \) m. \( A = \pi r^2 = 3.14 \times (5)^2 = 3.14 \times 25 = 78.5 \) \( m^2 \). 5. **Chord Property:** * A chord of length 16 cm is drawn in a circle with a radius of 10 cm. Find the distance of the chord from the center. Let the chord be AB. Let M be the midpoint of AB. A radius drawn to A forms a right-angled triangle OMA, where OA is the hypotenuse (radius = 10 cm), AM is half the chord (\( 16/2 = 8 \) cm), and OM is the distance from the center. Using Pythagoras theorem: \( OM^2 + AM^2 = OA^2 \) \( OM^2 + 8^2 = 10^2 \) \( OM^2 + 64 = 100 \) \( OM^2 = 100 - 64 = 36 \) \( OM = \sqrt{36} = 6 \) cm. ### Exercises 1. Define the following terms: a) Radius b) Chord c) Tangent d) Sector 2. A circular garden has a diameter of 28 meters. Calculate its circumference and area. (Use \( \pi = \frac{22}{7} \)) 3. If the circumference of a circle is 62.8 cm, what is its radius? (Use \( \pi = 3.14 \)) 4. Draw a circle with center O and label: a) A radius OA b) A diameter BC c) A chord DE d) A tangent FG touching the circle at point H e) A minor arc CD 5. A chord is 24 cm long, and its perpendicular distance from the center of the circle is 5 cm. Find the radius of the circle. ### Assignment 1. A wheel has a radius of 35 cm. How many revolutions will it make to cover a distance of 330 meters? (Use \( \pi = \frac{22}{7} \)) 2. The area of a circular swimming pool is 314 \( m^2 \). What is the perimeter (circumference) of the pool? (Use \( \pi = 3.14 \)) 3. Explain the difference between a sector and a segment of a circle. You may use diagrams to illustrate your explanation. 4. Two concentric circles (circles with the same center) have radii of 8 cm and 15 cm respectively. Find the area of the region between the two circles (the annulus). ### Summary In this lesson, we laid the groundwork for understanding circles by exploring their fundamental vocabulary and properties. We learned about the center, radius, diameter, chords, arcs, sectors, segments, tangents, and secants. We also reviewed the crucial formulas for calculating the circumference (\( C = 2\pi r \)) and area (\( A = \pi r^2 \)) of a circle. Mastering these terms and concepts is essential as we move forward to explore more complex circle theorems and their applications. ## Area of a circle and circumference ### Introduction Welcome to this lesson on circles! Circles are fundamental geometric shapes that you encounter every day, from the wheels of a bicycle to the face of a clock. In this lesson, we will explore two important measurements related to circles: their circumference (the distance around them) and their area (the space they cover). Understanding these concepts is crucial for various applications in mathematics and real life. ### Key Concepts #### What is a Circle? A circle is a set of all points in a plane that are equidistant from a fixed central point. This fixed central point is called the **center** of the circle. * **Radius (r):** The distance from the center of the circle to any point on its boundary. All radii of a given circle have the same length. * **Diameter (d):** A straight line segment that passes through the center of the circle and has its endpoints on the circle's boundary. The diameter is twice the length of the radius. \[ d = 2r \] #### Circumference of a Circle The circumference of a circle is the perimeter or the total distance around the circle. It is a linear measurement. * **The Constant Pi (\( \pi \)):** The ratio of a circle's circumference to its diameter is a constant value, known as pi (\( \pi \)). It is an irrational number, meaning its decimal representation never ends and never repeats. Common approximations for \( \pi \) are 3.14 or \( \frac{22}{7} \). \[ \pi = \frac{\text{Circumference}}{\text{Diameter}} \] * **Formula for Circumference:** Since \( C/d = \pi \), we can rearrange the formula to find the circumference: \[ C = \pi d \] Since \( d = 2r \), we can also write the formula in terms of the radius: \[ C = 2 \pi r \] #### Area of a Circle The area of a circle is the amount of surface enclosed within its boundary. It is a measurement of two-dimensional space. * **Formula for Area:** The area of a circle depends on its radius and the constant \( \pi \). \[ A = \pi r^2 \] If you are given the diameter, remember to first calculate the radius (\( r = d/2 \)) before using the formula. ### Vocabulary * **Circle:** A round plane figure whose boundary consists of points equidistant from a fixed center point. * **Radius (r):** The distance from the center of a circle to any point on its circumference. * **Diameter (d):** A straight line segment that passes through the center of a circle and whose endpoints lie on the circle. It is twice the length of the radius (\( d = 2r \)). * **Circumference (C):** The perimeter or the distance around a circle. * **Area (A):** The amount of surface enclosed by a circle. * **Pi (\( \pi \)):** A mathematical constant representing the ratio of a circle's circumference to its diameter, approximately 3.142 or \( \frac{22}{7} \). ### Examples #### Example 1: Calculating Circumference (using radius) Calculate the circumference of a circle with a radius of 7 cm. (Use \( \pi = \frac{22}{7} \)) * **Solution:** Given: \( r = 7 \) cm Formula: \( C = 2 \pi r \) \[ C = 2 \times \frac{22}{7} \times 7 \] \[ C = 2 \times 22 \] \[ C = 44 \text{ cm} \] #### Example 2: Calculating Circumference (using diameter) A circular table has a diameter of 1.2 m. Find its circumference. (Use \( \pi = 3.14 \)) * **Solution:** Given: \( d = 1.2 \) m Formula: \( C = \pi d \) \[ C = 3.14 \times 1.2 \] \[ C = 3.768 \text{ m} \] #### Example 3: Calculating Area (using radius) Find the area of a circular garden with a radius of 14 m. (Use \( \pi = \frac{22}{7} \)) * **Solution:** Given: \( r = 14 \) m Formula: \( A = \pi r^2 \) \[ A = \frac{22}{7} \times 14^2 \] \[ A = \frac{22}{7} \times 196 \] \[ A = 22 \times 28 \] \[ A = 616 \text{ m}^2 \] #### Example 4: Calculating Area (using diameter) A circular pizza has a diameter of 30 cm. Calculate its area. (Use \( \pi = 3.14 \)) * **Solution:** Given: \( d = 30 \) cm First, find the radius: \( r = d/2 = 30/2 = 15 \) cm Formula: \( A = \pi r^2 \) \[ A = 3.14 \times 15^2 \] \[ A = 3.14 \times 225 \] \[ A = 706.5 \text{ cm}^2 \] #### Example 5: Finding Radius from Circumference A circle has a circumference of 62.8 cm. Find its radius. (Use \( \pi = 3.14 \)) * **Solution:** Given: \( C = 62.8 \) cm Formula: \( C = 2 \pi r \) \[ 62.8 = 2 \times 3.14 \times r \] \[ 62.8 = 6.28 r \] \[ r = \frac{62.8}{6.28} \] \[ r = 10 \text{ cm} \] ### Exercises 1. Calculate the circumference of a circle with: * a) Radius = 3.5 cm (Use \( \pi = \frac{22}{7} \)) * b) Diameter = 20 m (Use \( \pi = 3.14 \)) 2. Find the area of a circle with: * a) Radius = 5 cm (Use \( \pi = 3.14 \)) * b) Diameter = 28 m (Use \( \pi = \frac{22}{7} \)) 3. A bicycle wheel has a diameter of 70 cm. How far does the bicycle travel if the wheel makes 100 revolutions? (Use \( \pi = \frac{22}{7} \)) 4. The area of a circular disc is 154 cm\( ^2 \). Find its radius. (Use \( \pi = \frac{22}{7} \)) ### Assignment 1. A circular rug has a radius of 1.5 meters. * a) Calculate its circumference. * b) Calculate the area it covers. (Use \( \pi = 3.14 \), round to two decimal places). 2. A circular track has a circumference of 440 meters. * a) Find the radius of the track. * b) What is the area enclosed by the track? (Use \( \pi = \frac{22}{7} \)). 3. A circular plate has an area of 201.06 cm\( ^2 \). Calculate its diameter. (Use \( \pi = 3.14 \)). ### Summary In this lesson, we have learned how to calculate the circumference and area of a circle. Remember the key formulas: * **Circumference (C):** The distance around the circle. * \( C = \pi d \) or \( C = 2 \pi r \) * **Area (A):** The space covered by the circle. * \( A = \pi r^2 \) Always identify whether you are given the radius (r) or diameter (d) and choose the appropriate formula. Pay attention to the value of \( \pi \) you are instructed to use and ensure your units are correct (linear units for circumference, square units for area). ## Arc length, Area of a sector and Area of a segment. ### Introduction Welcome to this lesson on Circles and Circle Theorems! Today, we will dive into calculating specific parts of a circle: the length of an arc, the area of a sector, and the area of a segment. These concepts are crucial for understanding how to measure parts of circular shapes and have many practical applications in engineering, design, and everyday life. ### Key Concepts #### 1. Arc Length An **arc** is a portion of the circumference of a circle. Imagine cutting a piece of the circle's edge – that's an arc! The length of an arc depends on the radius of the circle and the angle the arc subtends at the center of the circle. To calculate the length of an arc, \(L\), when the angle \(\theta\) is given in degrees: \[ L = \frac{\theta}{360^\circ} \times 2\pi r \] Where: * \(L\) is the arc length. * \(\theta\) is the angle subtended by the arc at the center of the circle, measured in degrees. * \(r\) is the radius of the circle. * \(2\pi r\) is the total circumference of the circle. #### 2. Area of a Sector A **sector** of a circle is the region bounded by two radii and the arc connecting their endpoints. Think of it like a slice of pizza or a piece of pie. Its area depends on the radius of the circle and the angle of the sector. To calculate the area of a sector, \(A_{sector}\), when the angle \(\theta\) is given in degrees: \[ A_{sector} = \frac{\theta}{360^\circ} \times \pi r^2 \] Where: * \(A_{sector}\) is the area of the sector. * \(\theta\) is the angle subtended by the sector at the center of the circle, measured in degrees. * \(r\) is the radius of the circle. * \(\pi r^2\) is the total area of the circle. #### 3. Area of a Segment A **segment** of a circle is the region bounded by a chord and the arc it cuts off. It's the area left over when you cut a triangular piece from a sector. To find the area of a segment, you first find the area of the sector and then subtract the area of the triangle formed by the two radii and the chord. To calculate the area of a segment, \(A_{segment}\): \[ A_{segment} = A_{sector} - A_{triangle} \] The triangle formed by the two radii and the chord is an isosceles triangle with two sides equal to the radius \(r\) and the angle between them equal to \(\theta\). The area of this triangle, \(A_{triangle}\), can be found using the formula: \[ A_{triangle} = \frac{1}{2} r^2 \sin\theta \] Therefore, the formula for the area of a segment is: \[ A_{segment} = \left( \frac{\theta}{360^\circ} \times \pi r^2 \right) - \left( \frac{1}{2} r^2 \sin\theta \right) \] Where: * \(A_{segment}\) is the area of the segment. * \(A_{sector}\) is the area of the corresponding sector. * \(A_{triangle}\) is the area of the triangle formed by the two radii and the chord. * \(\theta\) is the angle subtended at the center of the circle, in degrees. * \(r\) is the radius of the circle. ### Vocabulary * **Arc:** A continuous portion of the circumference of a circle. * **Sector:** A region of a circle bounded by two radii and the arc between them. * **Segment:** A region of a circle bounded by a chord and the arc subtended by the chord. * **Radius (r):** The distance from the center of a circle to any point on its circumference. * **Circumference:** The perimeter or distance around a circle. * **Chord:** A straight line segment connecting two points on the circumference of a circle. * **Angle Subtended at the Center (\(\theta\)):** The angle formed at the center of the circle by two radii or by radii to the endpoints of an arc or chord. ### Examples (Use \( \pi = 3.142 \) and give answers to 3 significant figures where appropriate) #### Example 1: Arc Length A circle has a radius of 10 cm. Find the length of an arc that subtends an angle of \(72^\circ\) at the center. **Solution:** Given: \(r = 10 \text{ cm}\), \(\theta = 72^\circ\) Formula: \[ L = \frac{\theta}{360^\circ} \times 2\pi r \] Substitute values: \[ L = \frac{72}{360} \times 2 \times 3.142 \times 10 \] \[ L = \frac{1}{5} \times 62.84 \] \[ L = 12.568 \text{ cm} \] Rounding to 3 s.f., \(L = 12.6 \text{ cm}\). #### Example 2: Area of a Sector Calculate the area of a sector with a radius of 7 cm and a central angle of \(120^\circ\). **Solution:** Given: \(r = 7 \text{ cm}\), \(\theta = 120^\circ\) Formula: \[ A_{sector} = \frac{\theta}{360^\circ} \times \pi r^2 \] Substitute values: \[ A_{sector} = \frac{120}{360} \times 3.142 \times (7)^2 \] \[ A_{sector} = \frac{1}{3} \times 3.142 \times 49 \] \[ A_{sector} = \frac{1}{3} \times 153.958 \] \[ A_{sector} = 51.319 \text{ cm}^2 \] Rounding to 3 s.f., \(A_{sector} = 51.3 \text{ cm}^2\). #### Example 3: Area of a Segment Find the area of the minor segment of a circle with a radius of 8 cm and a central angle of \(90^\circ\). **Solution:** Given: \(r = 8 \text{ cm}\), \(\theta = 90^\circ\) **Step 1: Calculate the Area of the Sector** Formula: \[ A_{sector} = \frac{\theta}{360^\circ} \times \pi r^2 \] Substitute values: \[ A_{sector} = \frac{90}{360} \times 3.142 \times (8)^2 \] \[ A_{sector} = \frac{1}{4} \times 3.142 \times 64 \] \[ A_{sector} = 1 \times 3.142 \times 16 \] \[ A_{sector} = 50.272 \text{ cm}^2 \] **Step 2: Calculate the Area of the Triangle** Formula: \[ A_{triangle} = \frac{1}{2} r^2 \sin\theta \] Substitute values: \[ A_{triangle} = \frac{1}{2} (8)^2 \sin(90^\circ) \] Since \(\sin(90^\circ) = 1\): \[ A_{triangle} = \frac{1}{2} \times 64 \times 1 \] \[ A_{triangle} = 32 \text{ cm}^2 \] **Step 3: Calculate the Area of the Segment** Formula: \[ A_{segment} = A_{sector} - A_{triangle} \] Substitute values: \[ A_{segment} = 50.272 - 32 \] \[ A_{segment} = 18.272 \text{ cm}^2 \] Rounding to 3 s.f., \(A_{segment} = 18.3 \text{ cm}^2\). ### Exercises (Use \( \pi = 3.142 \) unless otherwise stated. Give answers to 3 significant figures where appropriate) 1. A wheel has a radius of 20 cm. What is the length of the arc covered by a spoke that turns through an angle of \(100^\circ\)? 2. Find the area of a sector of a circle with a radius of 15 cm and a central angle of \(60^\circ\). 3. The area of a sector is \(45 \text{ cm}^2\) and its radius is 9 cm. Calculate the angle subtended at the center. 4. A circle has a radius of 12 cm. Find the area of the minor segment cut off by a chord that subtends an angle of \(150^\circ\) at the center. 5. An arc of a circle is 25 cm long and subtends an angle of \(180^\circ\) at the center. Calculate the radius of the circle. ### Assignment (Use \( \pi = 3.142 \) unless otherwise stated. Give answers to 3 significant figures where appropriate) 1. A pendulum of length 50 cm swings through an angle of \(15^\circ\). What is the length of the arc traced by the bob? 2. A circular garden has a radius of 8 meters. A sprinkler covers a sector of the garden with an angle of \(135^\circ\). Calculate the area of the garden watered by the sprinkler. 3. The area of a major sector of a circle with radius 10 cm is \(250 \text{ cm}^2\). Find the angle of the major sector and the length of its arc. 4. A chord of a circle of radius 14 cm subtends an angle of \(70^\circ\) at the center. Calculate the area of the minor segment. 5. A sector of a circle has an area of \(77 \text{ cm}^2\) and a radius of 7 cm. Find: a. The angle of the sector. b. The perimeter of the sector (i.e., the sum of the arc length and two radii). ### Summary In this lesson, we explored how to calculate three important measurements related to parts of a circle: * **Arc Length:** A portion of the circle's circumference, calculated using the formula \( L = \frac{\theta}{360^\circ} \times 2\pi r \). * **Area of a Sector:** A pie-slice portion of the circle's area, calculated using the formula \( A_{sector} = \frac{\theta}{360^\circ} \times \pi r^2 \). * **Area of a Segment:** The area bounded by a chord and an arc, found by subtracting the area of the triangle formed by the radii and the chord from the area of the corresponding sector: \( A_{segment} = A_{sector} - \frac{1}{2} r^2 \sin\theta \). Remember that the central angle \(\theta\) and the radius \(r\) are key components in all these calculations. Mastering these formulas will help you solve a variety of geometry problems involving circular shapes. ## Circle theorems : Angle properties ### Introduction Welcome to this lesson on Circle Theorems, specifically focusing on the angle properties of circles. Circles are fundamental shapes in mathematics, and understanding their properties helps us solve complex geometric problems. In this lesson, we will explore several key theorems that define the relationships between angles formed by chords, tangents, and arcs within a circle. These theorems are powerful tools for calculating unknown angles and proving geometric relationships. ### Key Concepts #### 1. Angle at the Centre and Circumference * **Theorem:** The angle subtended by an arc at the centre of a circle is twice the angle subtended by the same arc at any point on the remaining part of the circumference. * **Explanation:** Imagine an arc AB. If you draw lines from A and B to the centre O, you form \( \angle AOB \). If you draw lines from A and B to any point C on the circumference, you form \( \angle ACB \). This theorem states that \[ \angle AOB = 2 \times \angle ACB \] * **Note:** This applies whether C is on the major or minor arc, as long as it's the *remaining* part of the circumference relative to the arc defining the central angle. #### 2. Angles in the Same Segment * **Theorem:** Angles subtended by the same arc in the same segment of a circle are equal. * **Explanation:** If you have an arc AB and two points C and D on the circumference in the same segment (e.g., both on the major arc), then \( \angle ACB \) and \( \angle ADB \) will be equal. \[ \angle ACB = \angle ADB \] #### 3. Angle in a Semicircle * **Theorem:** The angle subtended by a diameter at any point on the circumference is a right angle (90 degrees). * **Explanation:** This is a special case of the first theorem. If the arc is a semicircle (meaning it's subtended by a diameter), the angle at the centre is \( 180^ \) (a straight line). Therefore, the angle at the circumference will be half of \( 180^ \), which is \( 90^ \). * **Formula:** If AB is a diameter, and C is any point on the circumference, then \[ \angle ACB = 90^ \ ] #### 4. Cyclic Quadrilaterals * **Definition:** A cyclic quadrilateral is a quadrilateral whose four vertices all lie on the circumference of a circle. * **Theorem 1:** The opposite angles of a cyclic quadrilateral are supplementary (they add up to \( 180^ \)). * **Explanation:** For a cyclic quadrilateral ABCD, \[ \angle A + \angle C = 180^ \ ] and \[ \angle B + \angle D = 180^ \ ] * **Theorem 2 (Exterior Angle Property):** The exterior angle of a cyclic quadrilateral is equal to the interior opposite angle. * **Explanation:** If one side of a cyclic quadrilateral is extended, the exterior angle formed is equal to the interior angle at the opposite vertex. #### 5. Tangent-Chord Theorem (Alternate Segment Theorem) * **Definition:** A tangent is a line that touches a circle at exactly one point (the point of contact). A chord is a line segment connecting two points on the circumference. * **Theorem:** The angle between a tangent to a circle and a chord drawn from the point of contact is equal to the angle in the alternate segment. * **Explanation:** If a tangent AB touches the circle at point T, and TC is a chord from T, then the angle formed by the tangent and the chord (e.g., \( \angle AT C \)) is equal to the angle subtended by the chord in the alternate segment (e.g., \( \angle TBC \)). \[ \angle ATC = \angle TBC \] and \[ \angle BT D = \angle TAD \] (where D is a point in the other segment). ### Vocabulary * **Circle:** A set of all points in a plane that are equidistant from a central point. * **Centre:** The central point from which all points on the circle are equidistant. * **Radius (r):** A line segment from the centre to any point on the circumference. * **Diameter (d):** A chord that passes through the centre of the circle. It is the longest chord and \( d = 2r \). * **Chord:** A line segment connecting two points on the circumference. * **Arc:** A continuous portion of the circumference of a circle. * **Segment:** The region of a circle bounded by a chord and an arc. (Minor segment and Major segment). * **Tangent:** A line that touches the circle at exactly one point. * **Circumference:** The perimeter or boundary of a circle. * **Cyclic Quadrilateral:** A quadrilateral whose all four vertices lie on the circumference of a circle. * **Subtend:** To form an angle at a particular point. * **Alternate Segment:** The segment of a circle that does not contain the angle between a tangent and a chord at the point of contact. ### Examples #### Example 1: Angle at Centre and Circumference **Problem:** In a circle with centre O, points A, B, and C are on the circumference. If \( \angle AOB = 130^ \ ), find \( \angle ACB \). **Solution:** According to the theorem, the angle at the centre is twice the angle at the circumference subtended by the same arc. Therefore, \[ \angle AOB = 2 \times \angle ACB \] Given \( \angle AOB = 130^ \ ), So, \[ 130^ \ = 2 \times \angle ACB \] Divide both sides by 2: \[ \angle ACB = \frac{130^ \ }{2} \] \[ \angle ACB = 65^ \ ] #### Example 2: Angles in the Same Segment **Problem:** Points A, B, C, and D are on the circumference of a circle. If \( \angle ADB = 40^ \ ) and \( \angle DBC = 35^ \ ), find \( \angle ACB \) and \( \angle DAC \). **Solution:** 1. **Find \( \angle ACB \):** Angles subtended by the same arc are equal. Arc AB subtends \( \angle ADB \) and \( \angle ACB \). Therefore, \[ \angle ACB = \angle ADB \] \[ \angle ACB = 40^ \ ] 2. **Find \( \angle DAC \):** Arc DC subtends \( \angle DBC \) and \( \angle DAC \). Therefore, \[ \angle DAC = \angle DBC \] \[ \angle DAC = 35^ \ ] #### Example 3: Angle in a Semicircle **Problem:** AB is the diameter of a circle with centre O. Point C is on the circumference. If \( \angle BAC = 30^ \ ), find \( \angle ABC \). **Solution:** Since AB is the diameter, the angle subtended by the diameter at any point on the circumference is a right angle. Therefore, \[ \angle ACB = 90^ \ ] Now, consider \( \triangle ABC \). The sum of angles in a triangle is \( 180^ \ ). \[ \angle BAC + \angle ABC + \angle ACB = 180^ \ ] \[ 30^ \ + \angle ABC + 90^ \ = 180^ \ ] \[ 120^ \ + \angle ABC = 180^ \ ] \[ \angle ABC = 180^ \ - 120^ \ ] \[ \angle ABC = 60^ \ ] #### Example 4: Cyclic Quadrilateral **Problem:** In a cyclic quadrilateral ABCD, \( \angle A = 85^ \ ) and \( \angle B = 70^ \ ). Find \( \angle C \) and \( \angle D \). **Solution:** In a cyclic quadrilateral, opposite angles are supplementary (add up to \( 180^ \ )). 1. **Find \( \angle C \):** \( \angle A \) and \( \angle C \) are opposite angles. \[ \angle A + \angle C = 180^ \ ] \[ 85^ \ + \angle C = 180^ \ ] \[ \angle C = 180^ \ - 85^ \ ] \[ \angle C = 95^ \ ] 2. **Find \( \angle D \):** \( \angle B \) and \( \angle D \) are opposite angles. \[ \angle B + \angle D = 180^ \ ] \[ 70^ \ + \angle D = 180^ \ ] \[ \angle D = 180^ \ - 70^ \ ] \[ \angle D = 110^ \ ] #### Example 5: Tangent-Chord Theorem **Problem:** A tangent line XY touches a circle at point P. PQ is a chord. If \( \angle QPX = 70^ \ ), find \( \angle PRQ \), where R is a point on the circumference in the alternate segment. **Solution:** According to the Tangent-Chord Theorem, the angle between the tangent and the chord is equal to the angle in the alternate segment. Given \( \angle QPX = 70^ \ ), which is the angle between tangent XY and chord PQ. Therefore, the angle subtended by chord PQ in the alternate segment, \( \angle PRQ \), is equal to \( \angle QPX \). \[ \angle PRQ = \angle QPX \] \[ \angle PRQ = 70^ \ ] ### Exercises 1. In a circle with centre O, points P, Q, R are on the circumference. If \( \angle POR = 150^ \ ), find \( \angle PQR \). 2. A, B, C, D are points on a circle. If \( \angle BAC = 55^ \ ) and \( \angle ABD = 60^ \ ), find \( \angle BDC \) and \( \angle ACD \). 3. MN is the diameter of a circle. Point K is on the circumference. If \( \angle MNK = 48^ \ ), calculate \( \angle MKN \) and \( \angle KMN \). 4. PQRS is a cyclic quadrilateral. If \( \angle P = 100^ \ ) and \( \angle Q = 75^ \ ), find \( \angle R \) and \( \angle S \). 5. Line AB is tangent to a circle at point C. Chord CD is drawn. If \( \angle ACD = 62^ \ ), find the angle subtended by chord CD in the alternate segment. ### Assignment 1. In the figure below, O is the centre of the circle. A, B, C, D are points on the circumference. \( \angle AOC = 140^ \ ) and \( \angle ADB = 30^ \ ). Find: a) \( \angle ABC \) b) \( \angle ADC \) c) \( \angle BAC \) (Hint: Draw a diagram for better visualization. Use the properties you've learned to find the angles step-by-step.) 2. A cyclic quadrilateral ABCD has its diagonal AC bisecting \( \angle DAB \). If \( \angle ADC = 110^ \ ), find \( \angle ABC \) and \( \angle DAC \). 3. The tangent at point T on a circle is parallel to the chord AB. Prove that the triangle TAB is an isosceles triangle. ### Summary In this lesson, we have covered the essential angle properties related to circles: * The **angle at the centre** is twice the **angle at the circumference** subtended by the same arc. * **Angles in the same segment** subtended by the same arc are equal. * The **angle in a semicircle** is always \( 90^ \). * **Opposite angles of a cyclic quadrilateral** sum to \( 180^ \ ), and its exterior angle equals the interior opposite angle. * The **Tangent-Chord Theorem** states that the angle between a tangent and a chord from the point of contact is equal to the angle in the alternate segment. Mastering these theorems is crucial for solving problems involving circles. Remember to identify the specific theorem applicable to each situation and apply it systematically. Consistent practice will help solidify your understanding. ## Circle theorems : Intersecting chords ### Introduction Welcome to this lesson on circle theorems, focusing specifically on intersecting chords, secants, and tangents. In geometry, when lines or segments interact with circles, they create predictable relationships. Today, we will explore the powerful theorems that describe these relationships when chords, secants, and tangents intersect both inside and outside a circle. Understanding these theorems will allow you to solve for unknown lengths in various geometric problems. ### Key Concepts #### 1. Intersecting Chords Theorem (Inside the Circle) When two chords intersect *inside* a circle, the product of the segments of one chord is equal to the product of the segments of the other chord. This theorem provides a fundamental way to relate the lengths of segments created by intersecting chords. * **Illustration:** Imagine two chords, AB and CD, intersecting at a point P inside the circle. * **Formula:** If chords AB and CD intersect at point P, then: \[ AP \cdot PB = CP \cdot PD \] Where AP, PB, CP, and PD are the lengths of the segments created by the intersection. #### 2. Intersecting Secants Theorem (Outside the Circle) When two secant segments are drawn to a circle from an exterior point, the product of the external segment and the entire secant segment is equal for both secants. A secant is a line that intersects a circle at two points. * **Illustration:** Consider an external point P, from which two secants, PAB and PCD, are drawn to the circle. A and C are points closer to P, and B and D are points further from P on the circle. * **Formula:** If secants PAB and PCD intersect at an external point P, then: \[ PA \cdot PB = PC \cdot PD \] Where PA and PC are the lengths of the external segments, and PB and PD are the lengths of the entire secant segments (from P to the further intersection point on the circle). #### 3. Tangent-Secant Theorem (Outside the Circle) When a tangent segment and a secant segment are drawn to a circle from an exterior point, the square of the length of the tangent segment is equal to the product of the external secant segment and the entire secant segment. A tangent is a line that touches the circle at exactly one point. * **Illustration:** From an external point P, a tangent PT touches the circle at point T, and a secant PAB intersects the circle at points A and B. * **Formula:** If tangent PT and secant PAB intersect at an external point P, then: \[ PT^2 = PA \cdot PB \] Where PT is the length of the tangent segment, PA is the length of the external secant segment, and PB is the length of the entire secant segment. ### Vocabulary * **Chord:** A line segment whose endpoints lie on the circle. * **Secant:** A line that intersects a circle at two distinct points. * **Tangent:** A line in the plane of a circle that intersects the circle at exactly one point, called the point of tangency. * **Intersection Point:** The point where two or more lines, segments, or arcs cross each other. * **Segment (of a chord/secant):** A part of a line segment. When chords intersect, they are divided into two segments. * **External Segment (of a secant):** The part of a secant segment that is outside the circle, from the external point to the first point of intersection with the circle. * **Entire Secant Segment:** The whole length of the secant segment from the external point to the second point of intersection with the circle. ### Examples #### Example 1: Intersecting Chords (Inside the Circle) Given two chords AB and CD intersecting at P inside a circle. If \( AP = 6 \) cm, \( PB = 4 \) cm, and \( CP = 3 \) cm, find the length of PD. * **Solution:** Using the Intersecting Chords Theorem: \( AP \cdot PB = CP \cdot PD \) Substitute the given values: \( 6 \cdot 4 = 3 \cdot PD \) \( 24 = 3 \cdot PD \) Divide both sides by 3: \( PD = \frac{24}{3} \) \( PD = 8 \) cm #### Example 2: Intersecting Secants (Outside the Circle) From an external point P, two secants PAB and PCD are drawn. If \( PA = 5 \) cm, \( PB = 12 \) cm, and \( PC = 6 \) cm, find the length of PD. * **Solution:** Using the Intersecting Secants Theorem: \( PA \cdot PB = PC \cdot PD \) Substitute the given values: \( 5 \cdot 12 = 6 \cdot PD \) \( 60 = 6 \cdot PD \) Divide both sides by 6: \( PD = \frac{60}{6} \) \( PD = 10 \) cm #### Example 3: Tangent-Secant Theorem (Outside the Circle) From an external point P, a tangent PT is drawn to a circle, and a secant PAB is drawn. If \( PT = 8 \) cm and \( PA = 4 \) cm, find the length of PB. * **Solution:** Using the Tangent-Secant Theorem: \( PT^2 = PA \cdot PB \) Substitute the given values: \( 8^2 = 4 \cdot PB \) \( 64 = 4 \cdot PB \) Divide both sides by 4: \( PB = \frac{64}{4} \) \( PB = 16 \) cm ### Exercises 1. Chords EF and GH intersect at point K inside a circle. If \( EK = 5 \) cm, \( KF = 8 \) cm, and \( GK = 10 \) cm, find the length of KH. 2. Secants PQR and PST are drawn from an external point P. If \( PQ = 4 \) cm, \( PR = 10 \) cm, and \( PS = 5 \) cm, find the length of PT. 3. A tangent MN touches a circle at N, and a secant MPQ intersects the circle at P and Q. If \( MN = 12 \) cm and \( MP = 9 \) cm, find the length of MQ. 4. Inside a circle, two chords intersect. The segments of one chord are 7 cm and \( x \) cm. The segments of the other chord are 6 cm and 14 cm. Find the value of \( x \). ### Assignment 1. Two chords AB and CD intersect at P. \( AP = x \), \( PB = x + 2 \), \( CP = 3 \), \( PD = 8 \). Find the value of \( x \). 2. From an external point P, a tangent PT is drawn such that \( PT = 10 \) cm. A secant PAB is drawn such that \( PA = 4 \) cm. Find the length of AB. 3. Secants PQR and PST are drawn from an external point P. If \( PQ = 3 \) cm, \( QR = 7 \) cm, and \( PS = 4 \) cm, find the length of ST. 4. Consider a circle with two chords intersecting inside. One chord has segments of lengths 10 cm and \( (y+1) \) cm. The other chord has segments of lengths \( y \) cm and \( (y+4) \) cm. Find the value(s) of \( y \). ### Summary Today, we explored three fundamental theorems related to intersecting chords, secants, and tangents in circles: * **Intersecting Chords Theorem:** When two chords intersect inside a circle, the product of the segments of one chord equals the product of the segments of the other chord: \( AP \cdot PB = CP \cdot PD \). * **Intersecting Secants Theorem:** When two secants intersect outside a circle, the product of the external segment and the entire secant segment is equal for both secants: \( PA \cdot PB = PC \cdot PD \). * **Tangent-Secant Theorem:** When a tangent and a secant intersect outside a circle, the square of the tangent segment equals the product of the external secant segment and the entire secant segment: \( PT^2 = PA \cdot PB \). These theorems are powerful tools for solving problems involving lengths of segments related to circles. ## Circle theorems : Tangents properties and chords ### Introduction Welcome to this lesson on circle theorems! In this topic, we will explore some fundamental properties related to tangents and chords of a circle. Understanding these theorems is crucial for solving various geometry problems involving circles. We will learn how lines that touch a circle (tangents) and lines that connect two points on a circle (chords) behave, and the special relationships they have with the circle's center and radius. ### Key Concepts #### Tangent Properties ##### What is a Tangent? A **tangent** is a straight line that touches a circle at exactly one point. This point is called the **point of tangency** or **point of contact**. ##### Property 1: Tangent-Radius Property The tangent to a circle is always perpendicular to the radius at the point of tangency. * If line AB is a tangent to a circle with center O at point P, then the radius OP is perpendicular to AB. * This means the angle formed between the radius and the tangent at the point of tangency is \( 90^ \) degrees. * Example: \[ \angle OPA = 90^ \] ##### Property 2: Tangents from an External Point If two tangents are drawn to a circle from an external point, then: 1. The lengths of the tangents from the external point to the points of tangency are equal. * If PA and PB are tangents from point P to a circle with center O, then \( PA = PB \). 2. The line segment joining the external point to the center of the circle (OP) bisects the angle between the tangents (\( \angle APB \)). * So, \( \angle APO = \angle BPO \). 3. The line segment joining the external point to the center of the circle (OP) bisects the angle between the radii to the points of tangency (\( \angle AOB \)). * So, \( \angle AOP = \angle BOP \). #### Chord Properties ##### What is a Chord? A **chord** is a straight line segment that connects two points on the circumference of a circle. The longest chord in a circle is its diameter. ##### Property 1: Perpendicular from Center to Chord 1. The perpendicular drawn from the center of a circle to a chord bisects the chord. * If OM is perpendicular to chord AB (where M is on AB), then M is the midpoint of AB, so \( AM = MB \). 2. Conversely, the line segment joining the center of a circle to the midpoint of a chord is perpendicular to the chord. ##### Property 2: Equal Chords 1. Equal chords are equidistant from the center of the circle. * If chord AB = chord CD, then their perpendicular distances from the center O are equal (e.g., \( OX = OY \)). 2. Conversely, chords that are equidistant from the center of a circle are equal in length. ##### Property 3: Chords and Arcs 1. Equal chords subtend equal angles at the center. 2. Equal chords cut off equal arcs. 3. Conversely, if two arcs are equal, then their corresponding chords are equal. #### Tangent-Chord Theorem (Alternate Segment Theorem) ##### What is the Alternate Segment Theorem? The angle between a tangent and a chord through the point of contact is equal to the angle in the alternate segment. * Consider a tangent AB touching a circle at point P, and PC is a chord through P. * If Q is a point on the circumference in the alternate segment (the segment that does not contain the angle \( \angle BPC \)), then \[ \angle BPC = \angle PQC \] * Similarly, if R is a point on the circumference in the segment alternate to \( \angle APC \), then \[ \angle APC = \angle PRC \] ### Vocabulary * **Circle:** A set of all points equidistant from a central point. * **Radius (r):** A line segment from the center to any point on the circumference. * **Diameter (d):** A chord passing through the center of the circle. It is twice the radius (\( d = 2r \)). * **Chord:** A line segment connecting two points on the circumference. * **Tangent:** A line that touches the circle at exactly one point. * **Point of Tangency:** The single point where a tangent touches a circle. * **Arc:** A part of the circumference of a circle. * **Segment:** The region of a circle bounded by a chord and an arc. * **Alternate Segment:** The segment of the circle not containing the angle between the tangent and the chord. * **Equidistant:** At the same distance. * **Bisect:** To divide into two equal parts. * **Perpendicular:** Forming a right angle (90 degrees). ### Examples #### Example 1: Tangent-Radius Property A tangent AB touches a circle with center O at point P. The radius of the circle is 5 cm. If the distance from the center O to point A on the tangent is 13 cm, find the length of AP. **Solution:** Since the tangent is perpendicular to the radius at the point of tangency, \( \triangle OPA \) is a right-angled triangle with the right angle at P. Using the Pythagorean theorem: \[ OA^2 = OP^2 + AP^2 \] We are given \( OA = 13 \) cm (hypotenuse) and \( OP = 5 \) cm (radius). \[ 13^2 = 5^2 + AP^2 \] \[ 169 = 25 + AP^2 \] \[ AP^2 = 169 - 25 \] \[ AP^2 = 144 \] \[ AP = \sqrt{144} \] \[ AP = 12 \text{ cm} \] #### Example 2: Tangents from an External Point From an external point P, tangents PA and PB are drawn to a circle with center O. If \( \angle APB = 70^ \), find \( \angle AOB \) and \( \angle APO \). **Solution:** We know that the radius is perpendicular to the tangent at the point of tangency, so \( \angle OAP = 90^ \) and \( \angle OBP = 90^ \). The sum of angles in quadrilateral OAPB is \( 360^ \). \[ \angle AOB + \angle OAP + \angle OBP + \angle APB = 360^ \] \[ \angle AOB + 90^ + 90^ + 70^ = 360^ \] \[ \angle AOB + 250^ = 360^ \] \[ \angle AOB = 360^ - 250^ \] \[ \angle AOB = 110^ \] Also, the line segment OP bisects \( \angle APB \). \[ \angle APO = \frac{\angle APB}{2} = \frac{70^ }{2} = 35^ \] #### Example 3: Perpendicular from Center to Chord A chord of a circle is 16 cm long. If the perpendicular distance from the center to the chord is 6 cm, find the radius of the circle. **Solution:** The perpendicular from the center to a chord bisects the chord. So, half the length of the chord is \( \frac{16}{2} = 8 \) cm. Let the radius be \( r \). We have a right-angled triangle formed by the radius (hypotenuse), half-chord, and the perpendicular distance from the center. Using the Pythagorean theorem: \[ r^2 = (\text{half-chord})^2 + (\text{perpendicular distance})^2 \] \[ r^2 = 8^2 + 6^2 \] \[ r^2 = 64 + 36 \] \[ r^2 = 100 \] \[ r = \sqrt{100} \] \[ r = 10 \text{ cm} \] #### Example 4: Alternate Segment Theorem Line AB is a tangent to a circle at point P. Chord PC is drawn. If the angle subtended by chord PC in the alternate segment is \( 65^ \) (i.e., \( \angle PQC = 65^ \)), find \( \angle BPC \). **Solution:** According to the Alternate Segment Theorem, the angle between the tangent (AB) and the chord (PC) at the point of contact (P) is equal to the angle in the alternate segment (\( \angle PQC \)). \[ \angle BPC = \angle PQC \] Given \( \angle PQC = 65^ \). Therefore, \[ \angle BPC = 65^ \] ### Exercises 1. A tangent of length 24 cm is drawn from an external point to a circle with radius 7 cm. Calculate the distance of the external point from the center of the circle. 2. Two tangents PA and PB are drawn from a point P to a circle with center O. If \( \angle AOP = 50^ \), find \( \angle APB \). 3. The radius of a circle is 13 cm. A chord is drawn at a distance of 5 cm from the center. Find the length of the chord. 4. In a circle, two parallel chords of lengths 6 cm and 8 cm are on opposite sides of the center. If the distance between them is 7 cm, find the radius of the circle. 5. In the diagram, AB is a tangent to the circle at T. CD is a chord. If \( \angle CTA = 70^ \), find \( \angle CDT \). ### Assignment 1. From an external point R, tangents RA and RB are drawn to a circle with center O. If the radius is 8 cm and \( OR = 17 \) cm, calculate the length of RA. 2. A chord of a circle is 24 cm long. The radius of the circle is 13 cm. Find the perpendicular distance of the chord from the center. 3. Tangents PQ and PR are drawn from an external point P to a circle with center O. If \( \angle QPR = 80^ \), what is the measure of \( \angle QOR \)? 4. Line XY is tangent to a circle at point C. Chord CD is drawn. If \( \angle DCY = 55^ \), find the angle subtended by chord CD in the alternate segment. 5. A circular pond has a diameter of 20 meters. A straight path AB of length 16 meters is built across the pond. How far is the path from the center of the pond? ### Summary In this lesson, we have explored key circle theorems related to tangents and chords. We learned that: * A tangent is perpendicular to the radius at the point of tangency. * Two tangents drawn from an external point to a circle are equal in length. * The perpendicular from the center of a circle to a chord bisects the chord. * Equal chords are equidistant from the center of the circle. * The Alternate Segment Theorem states that the angle between a tangent and a chord through the point of contact is equal to the angle in the alternate segment. These properties are fundamental in solving various geometrical problems and will be very useful in your further studies in mathematics. ## Cyclic quadrilaterals and properties, Concentric circles and rings ### Introduction Welcome to this lesson on advanced circle geometry! In this topic, we will explore special types of quadrilaterals called cyclic quadrilaterals, whose vertices all lie on a single circle. We will learn about their unique properties related to their angles. Additionally, we will delve into concentric circles, which are circles sharing the same center, and understand how to calculate the area of the region between them, known as a ring or annulus. ### Key Concepts #### Cyclic Quadrilaterals ##### Definition A **cyclic quadrilateral** is a quadrilateral whose four vertices all lie on the circumference of a single circle. The circle is said to circumscribe the quadrilateral. ##### Properties of a Cyclic Quadrilateral 1. **Opposite Angles Sum to \( 180^\circ \):** The sum of the measures of any pair of opposite angles in a cyclic quadrilateral is \( 180^\circ \). * If ABCD is a cyclic quadrilateral, then: * \( \angle A + \angle C = 180^\circ \) * \( \angle B + \angle D = 180^\circ \) 2. **Exterior Angle Property:** If one side of a cyclic quadrilateral is produced (extended), the exterior angle formed is equal to the interior opposite angle. * If side AB of cyclic quadrilateral ABCD is extended to a point E, then \( \angle CBE = \angle CDA \) (exterior angle at B equals the interior opposite angle at D). 3. **Converse Property 1:** If the sum of opposite angles of a quadrilateral is \( 180^\circ \), then the quadrilateral is cyclic. 4. **Converse Property 2:** If the exterior angle of a quadrilateral is equal to its interior opposite angle, then the quadrilateral is cyclic. #### Concentric Circles ##### Definition **Concentric circles** are two or more circles that share the same center point but have different radii. ##### Properties * They do not intersect each other. * They are parallel in the sense that the distance between their circumferences, measured along a radius, is constant (the difference in radii). * Examples include a target board or ripples formed when a stone is dropped into water. #### Rings (Annulus) ##### Definition A **ring**, also known as an **annulus**, is the region bounded by two concentric circles. It's the area between the inner and outer circles, resembling a washer or a doughnut shape. ##### Area of a Ring To find the area of a ring, we subtract the area of the inner (smaller) circle from the area of the outer (larger) circle. * Let \( R \) be the radius of the larger circle. * Let \( r \) be the radius of the smaller circle. * Area of larger circle \( = \pi R^2 \) * Area of smaller circle \( = \pi r^2 \) * Area of the Ring \( A = \text{Area of larger circle} - \text{Area of smaller circle} \) * Therefore, the formula for the area of a ring is: \[ A = \pi R^2 - \pi r^2 = \pi (R^2 - r^2) \] ### Vocabulary * **Cyclic Quadrilateral:** A quadrilateral whose all four vertices lie on the circumference of a circle. * **Concentric Circles:** Two or more circles that share the same center point but have different radii. * **Annulus/Ring:** The region bounded by two concentric circles. * **Circumference:** The perimeter of a circle. * **Radius (plural: radii):** The distance from the center of a circle to any point on its circumference. * **Exterior Angle:** An angle formed by one side of a polygon and the extension of an adjacent side. * **Interior Opposite Angle:** The angle inside a polygon that is opposite to a given exterior angle. ### Examples #### Example 1: Finding Angles in a Cyclic Quadrilateral **Problem:** In a cyclic quadrilateral ABCD, \( \angle A = 75^\circ \) and \( \angle B = 90^\circ \). Find the measures of \( \angle C \) and \( \angle D \). **Solution:** We use the property that opposite angles in a cyclic quadrilateral sum to \( 180^\circ \). * For \( \angle C \): \( \angle A + \angle C = 180^\circ \) \( 75^\circ + \angle C = 180^\circ \) \( \angle C = 180^\circ - 75^\circ \) \( \angle C = 105^\circ \) * For \( \angle D \): \( \angle B + \angle D = 180^\circ \) \( 90^\circ + \angle D = 180^\circ \) \( \angle D = 180^\circ - 90^\circ \) \( \angle D = 90^\circ \) #### Example 2: Applying the Exterior Angle Property **Problem:** In a cyclic quadrilateral PQRS, the side PQ is produced to a point T. If \( \angle QRS = 118^\circ \), find \( \angle SQT \). **Solution:** According to the exterior angle property of a cyclic quadrilateral, the exterior angle \( \angle SQT \) is equal to its interior opposite angle \( \angle QRS \). * Given \( \angle QRS = 118^\circ \) * Therefore, \( \angle SQT = 118^\circ \) #### Example 3: Calculating the Area of a Ring **Problem:** Two concentric circles have radii of 12 cm and 9 cm. Calculate the area of the ring between them. (Use \( \pi = 3.142 \)). **Solution:** * Radius of the larger circle, \( R = 12 \text{ cm} \) * Radius of the smaller circle, \( r = 9 \text{ cm} \) * Area of the ring \( A = \pi (R^2 - r^2) \) \( A = \pi (12^2 - 9^2) \) \( A = \pi (144 - 81) \) \( A = \pi (63) \) \( A = 3.142 \times 63 \) \( A = 197.946 \text{ cm}^2 \) ### Exercises 1. In a cyclic quadrilateral DEFG, \( \angle D = 100^\circ \) and \( \angle E = 70^\circ \). Find \( \angle F \) and \( \angle G \). 2. A quadrilateral MNOP has its vertices on a circle. If \( \angle MNO = 125^\circ \), find the measure of \( \angle MPO \). 3. The side XY of a cyclic quadrilateral WXYZ is produced to point A. If \( \angle WZY = 88^\circ \), what is the measure of \( \angle ZYA \)? 4. A circular path surrounds a fountain. The outer radius of the path is 8 meters and the inner radius is 6 meters. Calculate the area of the path. (Use \( \pi = 3.14 \)). 5. Determine whether a quadrilateral with angles \( 60^\circ, 120^\circ, 100^\circ, 80^\circ \) can be a cyclic quadrilateral. Explain your reasoning. ### Assignment 1. In the figure below, ABCD is a cyclic quadrilateral. \( \angle BCD \) is \( (3x + 10)^\circ \) and \( \angle BAD \) is \( (2x + 50)^\circ \). Find the value of \( x \) and the measures of \( \angle BCD \) and \( \angle BAD \). 2. A circular lawn has a gravel border around it. The diameter of the lawn (inner circle) is 20 m, and the width of the border is 2.5 m. Calculate the area of the gravel border. (Use \( \pi = 3.142 \)). 3. Prove that an isosceles trapezium is always a cyclic quadrilateral. 4. Consider a square ABCD. Is it a cyclic quadrilateral? If so, what can you say about the center of the circumscribing circle? ### Summary In this lesson, we explored cyclic quadrilaterals, which are quadrilaterals whose vertices lie on a circle. We learned two crucial properties: the sum of opposite angles is \( 180^\circ \), and the exterior angle is equal to the interior opposite angle. We also investigated concentric circles, which are circles sharing a common center but having different radii. Finally, we learned how to calculate the area of a ring (annulus), the region between two concentric circles, using the formula \[ A = \pi (R^2 - r^2) \]. These concepts are fundamental in understanding more complex geometric problems involving circles and polygons.