Lesson Worksheet: The Power of a Point Theorem Mathematics

In this worksheet, we will practice finding the power of a point with respect to a circle.

Q1:

A circle has center 𝑀 and radius 𝑟=21. Find the power of the point 𝐴 with respect to the circle given that 𝐴𝑀=25.

Q2:

A point is at a distance 40 from the center of a circle. If its power with respect to the circle is 81, what is the radius of the circle, rounded to the nearest integer?

Q3:

Determine the position of a point 𝐴 with respect to the circle 𝑁 if 𝑃(𝐴)=814.

  • Aoutside the circle
  • Binside the circle
  • Con the circumference of the circle

Q4:

The power of the points 𝐴, 𝐵, and 𝐶 with respect to the circle 𝐾 are 𝑃(𝐴)=4, 𝑃(𝐵)=14, and 𝑃(𝐶)=1. The circle 𝐾 has center 𝑀 and a radius of 10 cm. Calculate the distance between 𝑀 and each of the points.

  • A𝐴𝑀=226cm, 𝐵𝑀=114cm, 𝐶𝑀=311cm
  • B𝐴𝑀=104cm, 𝐵𝑀=114cm, 𝐶𝑀=99cm
  • C𝐴𝑀=14cm, 𝐵𝑀=26cm, 𝐶𝑀=3cm
  • D𝐴𝑀=14cm, 𝐵𝑀=24cm, 𝐶𝑀=9cm

Q5:

A circle with center 𝑀 and a point 𝐴 satisfy 𝑀𝐴=28cm and 𝑃(𝐴)=4. Using 𝜋=227, find the area and the circumference of the circle to the nearest integer.

  • AArea =88cm, circumference =176cm
  • BArea =2,451cm, circumference =176cm
  • CArea =2,451cm, circumference =88cm
  • DArea =4,903cm, circumference =88cm

Q6:

Two circles 𝑀 and 𝑁 intersect at points 𝐴 and 𝐵, and the point 𝐶 satisfies 𝐶𝐵𝐴 and 𝐶𝐵𝐴. 𝐷 and 𝐸 are the points where 𝐶𝐸 intersects the circle 𝑀 and 𝐶𝐹 is a tangent to 𝑁. Given that 𝐶𝐷=7 and 𝐷𝐸=12, find 𝑃(𝐶).

Q7:

A circle has two chords, 𝐴𝐶 and 𝐵𝐷, intersecting at 𝐸. Given that 𝐴𝐸𝐵𝐸=13 and 𝐶𝐸=6cm, find the length of 𝐷𝐸.

Q8:

A circle has two secants, 𝐴𝐵 and 𝐴𝐷, intersecting at 𝐴. Given that 𝐴𝐸=3cm, 𝐸𝐷=5cm, and 𝐴𝐵=9cm, find the length of 𝐵𝐶, giving your answer to the nearest tenth.

Q9:

A circle has a tangent 𝐴𝐵 and a secant 𝐴𝐷 that cut the circle at 𝐶. Given that 𝐴𝐵=7cm and 𝐴𝐶=5cm, find the length of 𝐶𝐷. Give your answer to the nearest hundredth.

Q10:

A circle with center 𝑀 has a radius of 11 cm. Point 𝐴 lies 8 cm from 𝑀 and belongs to the chord 𝐵𝐶. Given that 𝐴𝐵=3𝐴𝐶, calculate the length of 𝐵𝐶, giving your answer to the nearest hundredth.

This lesson includes 21 additional questions and 261 additional question variations for subscribers.

Nagwa uses cookies to ensure you get the best experience on our website. Learn more about our Privacy Policy.