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Worksheet: Power of a Point

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?

  • A41
  • B19
  • C21
  • D39

Q2:

A circle with center and a point satisfy and . Using as an approximation for , find the area and the circumference of the circle to the nearest integer.

  • Aarea , circumference
  • Barea , circumference
  • Carea , circumference
  • Darea , circumference

Q3:

Given that the point 𝐴 is outside the circle 𝑀 , and 𝐴 𝐷 is a tangent to the circle at 𝐷 such that 𝐴 𝐷 = 1 7 . 6 5 c m , find the power of the point 𝐴 with respect to the circle 𝑀 . Round your answer to the nearest hundredth.

Q4:

A circle has center 𝑀 and radius π‘Ÿ = 2 1 . Find the power of the point 𝐴 with respect to the circle given that 𝐴 𝑀 = 2 5 .

  • A4
  • B βˆ’ 1 8 4
  • C βˆ’ 4
  • D184

Q5:

A circle has center 𝑀 and radius π‘Ÿ = 4 5 . Find the power of the point 𝐴 with respect to the circle given that 𝐴 𝑀 = 4 4 .

  • A βˆ’ 1
  • B89
  • C1
  • D βˆ’ 8 9

Q6:

A circle has center 𝑀 and radius π‘Ÿ = 3 1 . Find the power of the point 𝐴 with respect to the circle given that 𝐴 𝑀 = 5 .

  • A βˆ’ 2 6
  • B936
  • C26
  • D βˆ’ 9 3 6

Q7:

Which of the following points lies on the circle of center ( 0 , 0 ) and radius 106?

  • A 𝐴 ( 9 8 , 9 8 )
  • B 𝐴 ( 9 0 , 9 0 )
  • C 𝐴 ( 1 8 , 4 7 )
  • D 𝐴 ( 9 0 , βˆ’ 5 6 )

Q8:

How many circles of radius 5.2 cm are there on points 𝐴 , 𝐡 with 𝐴 𝐡 = 2 4 c m ?

Q9:

Two circles 𝑀 and 𝑁 intersect at points 𝐴 and 𝐡 , and the point 𝐢 satisfies 𝐢 ∈  𝐡 𝐴 and 𝐢 βˆ‰ 𝐡 𝐴 . 𝐷 and 𝐸 are the points where 𝐢 𝐸 intersects the circle 𝑀 and οƒͺ 𝐢 𝐹 is a tangent to 𝑁 . Given that 𝐢 𝐷 = 7 and 𝐷 𝐸 = 1 2 , find 𝑃 ( 𝐢 )  .

Q10:

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

  • A 58.79 cm
  • B 21.91 cm
  • C 9.80 cm
  • D 26.29 cm

Q11:

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

  • A312
  • B βˆ’ 1 3 9 5
  • C 1 395
  • D βˆ’ 3 1 2

Q12:

How many points lie outside the circle?

  • A1
  • B2
  • C3
  • D7

Q13:

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

Q14:

A circle with center 𝑀 has a radius of 6 cm. The power of a point 𝐴 with respect to the circle is βˆ’ 3 5 . Decide whether 𝐴 is outside, inside, or on the circle and then find the distance between 𝐴 and 𝑀 .

  • Aon the circle, 41 cm
  • Boutside the circle, 29 cm
  • Cinside the circle, 1 cm

Q15:

Circle 𝑀 has diameter 6 cm. If 𝑀 𝐴 = 1 0 c m , what is the position of point 𝐴 with respect to the circle?

  • Aon
  • Binside
  • Cat the center of
  • Doutside

Q16:

Which points lie on the circle?

  • A 𝐻 , 𝐽
  • B 𝐴 , 𝐡 , 𝐢 , 𝐷 , 𝐸 , 𝐹 , 𝐺
  • C 𝐻 , 𝐽 , 𝐼
  • D 𝐼

Q17:

Which points lie outside the circle?

  • A 𝐡 , 𝐢 , 𝐸 , 𝐹 , 𝐻
  • B 𝐷 , 𝐽
  • C 𝐡 , 𝐢 , 𝐸 , 𝐹 , 𝐻 , 𝐷 , 𝐽
  • D 𝐴 , 𝐺 , 𝐼

Q18:

The power of the points 𝐴 , 𝐡 , and 𝐢 with respect to the circle 𝐾 are 𝑃 ( 𝐴 ) = 4 𝐾 , 𝑃 ( 𝐡 ) = 1 4 𝐾 , and 𝑃 ( 𝐢 ) = βˆ’ 1 𝐾 . 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 = 1 4 c m , 𝐡 𝑀 = 2 4 c m , 𝐢 𝑀 = 9 c m
  • B 𝐴 𝑀 = √ 1 4 c m , 𝐡 𝑀 = 2 √ 6 c m , 𝐢 𝑀 = 3 c m
  • C 𝐴 𝑀 = 1 0 4 c m , 𝐡 𝑀 = 1 1 4 c m , 𝐢 𝑀 = 9 9 c m
  • D 𝐴 𝑀 = 2 √ 2 6 c m , 𝐡 𝑀 = √ 1 1 4 c m , 𝐢 𝑀 = 3 √ 1 1 c m

Q19:

Determine the position of a point 𝐴 with respect to the circle 𝑁 if 𝑃 ( 𝐴 ) = 8 1 4 𝑁 .

  • Ainside the circle
  • Bon the circumference of the circle
  • Coutside the circle

Q20:

Circle 𝑀 has radius 3 cm. If 𝑀 𝐴 = 1 c m , what is the position of point 𝐴 with respect to the circle?

  • Aon
  • Boutside
  • Cinside

Q21:

The power of a point with respect to a circle is βˆ’ 5 7 5 when its distance from the center of that circle is 84. What is the circle’s diameter to the nearest hundredth?

  • A161.01
  • B87.36
  • C80.50
  • D174.71

Q22:

The power of a point with respect to a circle is 297 when its distance from the center of that circle is 94. What is the circle’s diameter to the nearest hundredth?

  • A191.13
  • B92.41
  • C95.57
  • D184.81

Q23:

How many circles can pass through two points?

  • A2
  • B1
  • C0
  • Dinfinite

Q24:

Which points lie outside the circle?

  • A 𝐴 , 𝐡 , 𝑋
  • B 𝑀 , 𝐷 , 𝐹
  • C 𝑄 , 𝐡 , 𝑀
  • D 𝐢 , 𝐸 , 𝑄

Q25:

How many circles can pass through three collinear points?

  • A2
  • B1
  • Cinfinite
  • D0