Question

Three circles, each of radius 20, have centres at P, Q, and R. Further, AB = 5, CD = 10 and EF = 12. What is the perimeter of \( \triangle PQR \)?

• A. 120
• B. 66
• C. 93
• D. 87

Hint:

For problems involving tangents between circles, remember to subtract the tangent lengths from the sum of the radii to find the distances between the centers of the circles.

Answer 1

The Correct Option is C

The three circles with centers at P, Q, and R each have a radius of 20. Let the distances between the centers of the circles be the sides of the triangle \( \triangle PQR \). We are given the following distances: - \( AB = 5 \), - \( CD = 10 \), - \( EF = 12 \). These segments are the lengths of the tangents from the points A, B, C, D, E, and F on the circles. Since the circles are tangent to each other, the sides of the triangle \( \triangle PQR \) are formed by adding the radii of two circles and subtracting the lengths of the tangents. Thus, we can calculate the perimeter of the triangle by adding the lengths of the sides \( PQ \), \( QR \), and \( PR \). Step 1: Calculate the side \( PQ \) The length of the side \( PQ \) is the sum of the radii minus the tangents between the circles. Since the radii of the circles are 20, and the distance between the centers is 20 + 20 = 40. Subtracting the tangent length \( AB = 5 \), we get: \[ PQ = 40 - 5 = 35. \] Step 2: Calculate the side \( QR \) Similarly, for the side \( QR \), we subtract the tangent length \( CD = 10 \) from the sum of the radii: \[ QR = 40 - 10 = 30. \] Step 3: Calculate the side \( PR \) For the side \( PR \), subtract the tangent length \( EF = 12 \) from the sum of the radii: \[ PR = 40 - 12 = 28. \] Step 4: Calculate the perimeter The perimeter of triangle \( \triangle PQR \) is the sum of the lengths of its sides: \[ \text{Perimeter} = PQ + QR + PR = 35 + 30 + 28 = 93. \]