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Circle Properties and Theorems

Grade 10 · Mathematics · Worksheet 2

  1. 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: ______________
  2. A circle with center (0,0) has radius 5. Find the length of the tangent from point (8,0) to the circle.
    Answer: ______________
  3. A circle with center (0,0) and radius 13 has a chord that is 24 units long. What is the perpendicular distance from the center of the circle to this chord?
    Answer: ______________
  4. A circle with center (0,0) has a chord from (3,4) to (6,0). Find the length of the chord. Answer: ______________
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Answer Key & Explanations

Circle Properties and Theorems · Grade 10 · Worksheet 2

  1. 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.

  2. A circle with center (0,0) has radius 5. Find the length of the tangent from point (8,0) to the circle. Answer: √39 Solution: We have a circle centered at (0,0) with radius 5. A tangent is drawn from the external point (8,0) to the circle. We need the length of this tangent.
    Full step-by-step solution

    Step 1: Understand the problem. We have a circle centered at (0,0) with radius 5. A tangent is drawn from the external point (8,0) to the circle. We need the length of this tangent. Step 2: Recall the geometric relationship. For a circle with center O and radius r, and an external point P, the tangent segment PT (where T is the point of tangency) is perpendicular to the radius OT. This forms a right triangle OTP with the right angle at T. Step 3: Identify the known lengths in this right triangle. - The radius OT is one leg. OT = 5. - The hypotenuse OP is the distance from the center O(0,0) to the external point P(8,0). - The tangent length PT is the other leg, which we need to find. Step 4: Calculate the distance OP. Coordinates: O(0,0) and P(8,0). Distance OP = sqrt( (8-0)^2 + (0-0)^2 ) = sqrt(64 + 0) = sqrt(64) = 8. So, OP = 8. Step 5: Apply the Pythagorean theorem to triangle OTP. In right triangle OTP: (OP)^2 = (OT)^2 + (PT)^2 Substitute known values: (8)^2 = (5)^2 + (PT)^2 64 = 25 + (PT)^2 Step 6: Solve for (PT)^2 and then PT. (PT)^2 = 64 - 25 (PT)^2 = 39 PT = sqrt(39) Step 7: State the final answer. The length of the tangent from point (8,0) to the circle is sqrt(39).

  3. A circle with center (0,0) and radius 13 has a chord that is 24 units long. What is the perpendicular distance from the center of the circle to this chord? Answer: 5 Solution: In a circle, the perpendicular from the center to a chord bisects the chord. So half the chord length is 24/2 = 12 units.
    Full step-by-step solution

    Step 1: In a circle, the perpendicular from the center to a chord bisects the chord. So half the chord length is 24/2 = 12 units. Step 2: The radius (13 units), half the chord length (12 units), and the perpendicular distance from the center form a right triangle, with the radius as the hypotenuse. Step 3: Using the Pythagorean theorem: (perpendicular distance)^2 + 12^2 = 13^2 Step 4: (perpendicular distance)^2 + 144 = 169 Step 5: (perpendicular distance)^2 = 169 - 144 = 25 Step 6: perpendicular distance = sqrt(25) = 5 The answer is 5.

  4. A circle with center (0,0) has a chord from (3,4) to (6,0). Find the length of the chord. Answer: 5 Solution: We have a circle centered at (0,0) with a chord from (3,4) to (6,0). We need the length of the chord.
    Full step-by-step solution

    Let's go step-by-step. --- **Step 1: Understand the problem** We have a circle centered at (0,0) with a chord from (3,4) to (6,0). We need the length of the chord. --- **Step 2: Recall the distance formula** The distance between two points (x1, y1) and (x2, y2) is: Distance = sqrt( (x2 - x1)^2 + (y2 - y1)^2 ) --- **Step 3: Apply the formula** Here, (x1, y1) = (3, 4) and (x2, y2) = (6, 0). x2 - x1 = 6 - 3 = 3 y2 - y1 = 0 - 4 = -4 --- **Step 4: Square the differences** (3)^2 = 9 (-4)^2 = 16 --- **Step 5: Sum and take square root** Sum = 9 + 16 = 25 Square root of 25 = 5 --- **Step 6: Conclusion** The chord length is 5. --- **Final Answer:** 5