Worksheet: The Power of a Point Theorem

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

Q1:

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?

Q2:

A circle with center 𝑀 and a point 𝐴 satisfy 𝑀𝐴=28cm and 𝑃(𝐴)=4. Using πœ‹=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

Q3:

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

Q4:

A circle with center 𝑁 has a diameter equal to 38 cm. A point 𝐡 satisfies 𝑁𝐡=7cm. Find the power of 𝐡 with respect to the circle, giving your answer to the nearest integer.

  • A βˆ’ 1 , 3 9 5
  • B312
  • C1,395
  • D βˆ’ 3 1 2

Q5:

A circle with center 𝑀 has a radius of 8 cm. The power of a point 𝐴 with respect to the circle is 36. Decide whether 𝐴 is outside, inside, or on the circle and then find the distance between 𝐴 and 𝑀.

  • AOn the circle, 28 cm
  • BInside the circle, 44 cm
  • COutside the circle, 10 cm

Q6:

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

  • A 𝐴 𝑀 = 2 √ 2 6 c m , 𝐡 𝑀 = √ 1 1 4 c m , 𝐢 𝑀 = 3 √ 1 1 c m
  • B 𝐴 𝑀 = 1 0 4 c m , 𝐡 𝑀 = 1 1 4 c m , 𝐢 𝑀 = 9 9 c m
  • C 𝐴 𝑀 = √ 1 4 c m , 𝐡 𝑀 = 2 √ 6 c m , 𝐢 𝑀 = 3 c m
  • D 𝐴 𝑀 = 1 4 c m , 𝐡 𝑀 = 2 4 c m , 𝐢 𝑀 = 9 c m

Q7:

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

Q8:

The power of a point with respect to a circle is βˆ’575 when its distance from the center of that circle is 84. What is the circle’s diameter to the nearest hundredth?

Q9:

Given that the point 𝐴 is outside the circle 𝑀, and 𝐴𝐷 is a tangent to the circle at 𝐷 such that 𝐴𝐷=17.65cm, find the power of the point 𝐴 with respect to the circle 𝑀. Round your answer to the nearest hundredth.

Q10:

A circle with center 𝑀 has a radius of 11 cm. Point 𝐴 lies 5 cm from 𝑀 and belongs to the chord 𝐡𝐢. Given that 𝐴𝐡=5𝐴𝐢, calculate 𝐡𝐢, giving your answer to the nearest hundredth.

Q11:

Two circles 𝑀 and 𝑁 intersect at points 𝐴 and 𝐡, and the point 𝐢 satisfies πΆβˆˆοƒ«π΅π΄ and πΆβˆ‰π΅π΄. 𝐷 and 𝐸 are the points where 𝐢𝐸 intersects the circle 𝑀 and οƒͺ𝐢𝐹 is a tangent to 𝑁. Given that 𝐢𝐷=7 and 𝐷𝐸=12, find 𝑃(𝐢).

Q12:

How many circles of radius 5.2 cm are there on points 𝐴, 𝐡 with 𝐴𝐡=24cm?

Q13:

A line 𝐿 intersects a circle with center 𝑀. Point 𝐴 lies on 𝐿 and is inside the circle. If the radius of the circle is 8 cm, π‘€π΄βŸ‚πΏ, and we set 𝑀𝐴=(3π‘₯βˆ’5)cm, in what interval does the value of π‘₯ belong?

  • A ο€Ό βˆ’ 5 3 , 1 3 3 
  • B ο€Ό 5 3 , 1 3 3 
  • C  βˆ’ 5 3 , 1 3 3 
  • D  5 3 , 1 3 3 

Q14:

A circle has a radius of 90 cm. A point lies on the circle at a distance of (3π‘₯βˆ’3) cm from the center. Which of the following is true?

  • A π‘₯ < 3 1
  • B π‘₯ > 3 1
  • C π‘₯ = 3 1

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