Circle Properties and Theorems

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Answer Key & Explanations

Circle Properties and Theorems · Grade 10 · Worksheet 2

1. Liam is designing a circular garden with a central fountain. The garden has a radius of 15 meters, and he wants to install a decorative stone path that forms a chord 24 meters long. What is the perpendicular distance from the center of the fountain to this chord? Answer: 9 meters Solution: We have a circular garden with radius \( R = 15 \) m. A chord of length \( 24 \) m is drawn in the circle. We want the perpendicular distance \( d \) from the center of the circle to the chord.
   Full step-by-step solution

   Let's go step-by-step. --- **Step 1: Understand the problem** We have a circular garden with radius \( R = 15 \) m. A chord of length \( 24 \) m is drawn in the circle. We want the perpendicular distance \( d \) from the center of the circle to the chord. --- **Step 2: Recall the geometry relationship** In a circle of radius \( R \), for a chord of length \( L \), the perpendicular distance \( d \) from the center to the chord is given by the right triangle: - Radius \( R \) is the hypotenuse. - Half the chord length \( L/2 \) is one leg. - The perpendicular distance \( d \) is the other leg. The formula from the Pythagorean theorem: \[ R^2 = d^2 + \left( \frac{L}{2} \right)^2 \] --- **Step 3: Substitute known values** \( R = 15 \) m \( L = 24 \) m So \( L/2 = 12 \) m \[ 15^2 = d^2 + 12^2 \] --- **Step 4: Calculate** \[ 225 = d^2 + 144 \] \[ d^2 = 225 - 144 \] \[ d^2 = 81 \] \[ d = \sqrt{81} = 9 \quad (\text{taking positive value since it's a distance}) \] --- **Step 5: Conclusion** The perpendicular distance from the center to the chord is \( 9 \) meters. --- **Final answer:** 9 meters

2. In a circle with center O, a chord AB is drawn. The perpendicular from O to AB meets AB at point M. If OM = 7 and the radius of the circle is 25, find the length of chord AB. Answer: 48 Solution: Let the radius be OA = 25. Since OM is perpendicular to chord AB, M is the midpoint of AB. Let AM = MB = x.
   Full step-by-step solution

   Step 1: Let the radius be OA = 25. Since OM is perpendicular to chord AB, M is the midpoint of AB. Let AM = MB = x. Step 2: In right triangle OMA, OA is the hypotenuse, OM = 7, and AM = x. By the Pythagorean theorem: x^2 + 7^2 = 25^2. Step 3: x^2 + 49 = 625. Step 4: x^2 = 625 - 49 = 576. Step 5: x = sqrt(576) = 24. Step 6: The full chord length AB = 2 * x = 2 * 24 = 48. The answer is 48.

3. A city is designing a circular park with a central fountain. The park has a radius of 25 meters. A straight walking path connects two points on the circular edge, and this path is located 7 meters from the center of the park at its closest point. What is the length of this walking path in meters? Answer: 48 Solution: In a circle, the perpendicular distance from the center to a chord, half the chord length, and the radius form a right triangle. The radius is 25 meters, and the perpendicular distance to the chord is 7 meters.
   Full step-by-step solution

   Step 1: In a circle, the perpendicular distance from the center to a chord, half the chord length, and the radius form a right triangle. Step 2: The radius is 25 meters, and the perpendicular distance to the chord is 7 meters. Step 3: Using the Pythagorean theorem: (half chord length)^2 + 7^2 = 25^2 Step 4: (half chord length)^2 + 49 = 625 Step 5: (half chord length)^2 = 625 - 49 = 576 Step 6: half chord length = sqrt(576) = 24 meters Step 7: Full chord length = 2 × 24 = 48 meters The answer is 48.

4. A circle with center at (0,0) has a chord from (3,4) to (3,-4). Find the length of the chord. Answer: 8 Solution: We have a circle centered at (0,0) and a chord from (3,4) to (3,-4). A chord is a line segment whose endpoints lie on the circle. Identify the coordinates of the endpoints.
   Full step-by-step solution

   Step 1: Understand the problem. We have a circle centered at (0,0) and a chord from (3,4) to (3,-4). A chord is a line segment whose endpoints lie on the circle. Step 2: Identify the coordinates of the endpoints. Point A = (3,4) Point B = (3,-4) Step 3: Find the length of the chord. The chord is vertical because both endpoints have the same x-coordinate (x = 3). The y-coordinates are 4 and -4. Step 4: Use the distance formula for vertical line segments. Length = |y2 - y1| = |(-4) - 4| = |-8| = 8. Step 5: Verify with the general distance formula. Distance = sqrt((3 - 3)^2 + (-4 - 4)^2) = sqrt(0 + (-8)^2) = sqrt(64) = 8. Step 6: Conclusion. The length of the chord is 8.

5. A circular fountain with radius 10 meters is being designed for a city park. The landscape architect wants to install a decorative stone path that forms a chord of the circle. This chord is located 6 meters from the center of the fountain. Additionally, two radii are drawn from the center to the endpoints of this chord, forming a central angle. What is the exact area of the circular sector bounded by these two radii and the arc between them? Answer: 100π/3 Solution: In circle geometry, the area of a sector depends on the central angle and the radius.
   Full step-by-step solution

   In circle geometry, the area of a sector depends on the central angle and the radius. When a chord is drawn at a specific distance from the center, it creates an isosceles triangle with the center and the chord endpoints. The relationship between the chord's distance from the center and the central angle can be found using right triangle trigonometry within this configuration.