Circle Properties and Theorems

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Answer Key & Explanations

Circle Properties and Theorems · Grade 10 · Worksheet 1

1. A circle has a radius of 15 cm. Two tangents are drawn from an external point P to the circle. The distance from P to the center of the circle is 17 cm. Find the length of each tangent segment. Answer: 8 cm Solution: Let O be the center of the circle, and let T be the point where one tangent touches the circle. The radius OT is perpendicular to the tangent PT, so triangle OTP is a right triangle with right angle at T. In triangle OTP, we know: OT = radius = 15 cm, OP = distance from P to center = 17 cm, and…
   Full step-by-step solution

   Step 1: Let O be the center of the circle, and let T be the point where one tangent touches the circle. The radius OT is perpendicular to the tangent PT, so triangle OTP is a right triangle with right angle at T. Step 2: In triangle OTP, we know: OT = radius = 15 cm, OP = distance from P to center = 17 cm, and PT is the tangent length we need to find. Step 3: Apply the Pythagorean theorem: PT^2 + OT^2 = OP^2 Step 4: Substitute the known values: PT^2 + 15^2 = 17^2 Step 5: Calculate: PT^2 + 225 = 289 Step 6: Solve for PT^2: PT^2 = 289 - 225 = 64 Step 7: Take the square root: PT = sqrt(64) = 8 cm The answer is 8 cm.

2. Noah is designing a circular fountain for a botanical garden. The fountain has a radius of 34 meters. He wants to install a straight stone path that connects two points on the circumference. The perpendicular distance from the center of the fountain to this chord is 16 meters. At the midpoint of the chord, Noah plans to place a small bronze plaque, and from that plaque, he will run a straight water pipe to the center of the fountain. What is the total length of the water pipe in meters? Answer: 16 Solution: The radius of the fountain is 34 meters. The perpendicular distance from the center to the chord is 16 meters.
   Full step-by-step solution

   Step 1: The radius of the fountain is 34 meters. The perpendicular distance from the center to the chord is 16 meters. Step 2: By the perpendicular bisector theorem, the line from the center of a circle to the midpoint of a chord is perpendicular to the chord. Step 3: The water pipe runs from the center of the fountain to the midpoint of the chord. Step 4: The length of this pipe is exactly the perpendicular distance from the center to the chord, which is 16 meters. The answer is 16.

3. A circular fountain with a radius of 10 meters is being designed for a city park. The landscape architect wants to create a triangular seating area inside the fountain where the vertices of the triangle lie on the circumference. If the triangle is equilateral, what is the exact area of this triangular seating area in square meters? Answer: 75*sqrt(3) Solution: For an equilateral triangle inscribed in a circle, the relationship between the side length (s) and the radius (R) is: s = R * sqrt(3) Given R = 10 meters, we calculate: s = 10 * sqrt(3) The area of an equilateral triangle is: Area = (s^2 * sqrt(3)) / 4 Substitute s = 10 * sqrt(3): Area = ((10 *…
   Full step-by-step solution

   Step 1: For an equilateral triangle inscribed in a circle, the relationship between the side length (s) and the radius (R) is: s = R * sqrt(3) Step 2: Given R = 10 meters, we calculate: s = 10 * sqrt(3) Step 3: The area of an equilateral triangle is: Area = (s^2 * sqrt(3)) / 4 Step 4: Substitute s = 10 * sqrt(3): Area = ((10 * sqrt(3))^2 * sqrt(3)) / 4 Step 5: Simplify: (100 * 3 * sqrt(3)) / 4 = (300 * sqrt(3)) / 4 Step 6: Reduce the fraction: 300/4 = 75, so Area = 75 * sqrt(3) The exact area of the triangular seating area is 75*sqrt(3) square meters.