Worksheet: Perpendicular Bisector of a Chord

In this worksheet, we will practice using the theory of the perpendicular bisector of a chord from the center of a circle and its converse to solve problems.


In the figure, circles 𝐽 and 𝐾 are congruent, 𝐴𝐵𝐶𝐷, 𝐴𝐵=(3𝑥+7)cm, and 𝐶𝐷=(8𝑥+12)cm. Find the length of 𝐴𝐵.


Points 𝑋 and 𝑌 are midpoints of segments 𝐴𝐵 and 𝐶𝐷 respectively. If 𝐴𝐵=60, what is 𝐶𝑌?


The radius of the circle 𝑂 below is 87 cm, 𝐴𝐵𝐶𝐷, and 𝑋 is the midpoint of 𝐴𝐵. Given that 𝑋𝑂 intersects 𝐶𝐷 at 𝑌 and 𝑂𝑌=60cm, find the length of 𝑌𝐶.


𝐴𝐵 is a chord in circle 𝑀 whose radius is 25.5 cm. If 𝐴𝐵=40.8cm, what is the length of 𝐷𝐸?


Given 𝐴𝑀=200cm and 𝑀𝐶=120cm, find 𝐴𝐵.


In circle 𝑀, suppose 𝑀𝐴=11 and 𝑀𝐶=9.4. Find the lengths of 𝐴𝐵 and 𝐶𝐷 to the nearest hundredth.

  • A5.71, 11.00
  • B11.43, 1.60
  • C22.00, 1.60
  • D11.43, 11.00


In the given circle, 𝑂𝐴=8.5 cm and 𝑂𝐶=4 cm. Determine the lengths of 𝐴𝐵 and 𝐶𝐷.

  • A𝐴𝐵=7.5cm, 𝐶𝐷=4cm
  • B𝐴𝐵=15cm, 𝐶𝐷=4cm
  • C𝐴𝐵=15 cm, 𝐶𝐷=4.5cm
  • D𝐴𝐵=7.5cm, 𝐶𝐷=4.5cm


The distance between the two lines 𝐴𝐷 and 𝐵𝐶 is 13 and 𝐴𝐷=𝐵𝐶. If the radius of the circle is 10.5, what is the length of 𝐴𝐷? Round your answer to two decimal places.


Given that 𝑀𝑋=𝑀𝑌, and 𝑌𝐷=31 cm, determine the length of 𝐴𝐵.


Given that 𝐴𝐵=𝐶𝐷, 𝐴𝐵=15cm, 𝑀𝐹=4𝑥cm, and 𝐶𝐷=(11𝑥+4)cm, determine the value of 𝑥 and the length of 𝐴𝑀.

  • A𝑥=4, 𝐴𝑀=72.25cm
  • B𝑥=1, 𝐴𝑀=7.5cm
  • C𝑥=1, 𝐴𝑀=8.5cm
  • D𝑥=4, 𝐴𝑀=15cm


Given that 𝐴𝐵=𝐶𝐷=(6𝑥+3)cm, 𝑀𝐸=(3𝑥+1)cm, and 𝑀𝑂=4cm, find the length of 𝐶𝐷.


In the circle 𝑀, if 𝐴𝐵=18.4cm, find the length of 𝐶𝐵.


In circle 𝑆, 𝑚𝑃𝑄𝑅=62. Find 𝑚𝑃𝑄.


The radius of circle 𝑀 is 60.9 cm, and 𝐴𝐵=84cm. What is the area of 𝐴𝐷𝐵?


In the figure, the two circles are concentric at 𝑀 and 𝐴𝐵=8. Calculate the area of the shaded region, to the nearest hundredth.


The circumference of circle 𝑀 is 36.6 cm. Calculate 𝐵𝐶 to the nearest tenth.


In the figure, the smaller of the concentric circles is tangent to chord 𝐴𝐵 at 𝐶. If the large circle has radius 67 cm, and 𝐴𝐵=120cm, what is the radius of the smaller circle?


If 𝑀𝐹>𝑀𝐸, find the range of values of 𝑥 that satisfies the data represented.

  • A(20,29)
  • B(4,24)
  • C(29,62)
  • D(20,62)


Point 𝑀 is at a distance 13 from 𝑂, which is the center of a circle of radius 7.3. Draw ray 𝑀𝐵 to intersect the circle at 𝐴 and 𝐵, with 𝐴 between 𝑀 and 𝐵. If 𝑀𝐴=5.7, what is 𝐴𝐵?


Given that 𝑀 and 𝑁 are the centers of two circles which intersect at 𝐴 and 𝐵 where 𝑀𝐴=18.7cm, 𝑁𝐴=24.8cm, and 𝑀𝑁=21.6cm, find the length of 𝐴𝐵. Round the answer to the nearest hundredth.


A circle has a radius of 48.9 cm. The point 𝐴 lies 48.3 cm from its center. If a chord 𝐵𝐶 satisfies 𝐴𝐵𝐶 and 𝐴𝐵=2𝐴𝐶, what is its length?


The radii of two concentric circles are 55 cm and 40 cm. 𝐴𝐷 is a chord in the larger circle, and 𝐴𝐷 intersects the smaller circle first at 𝐵 and then at 𝐶. Given that 𝐴𝐵=41cm, find the length of 𝐵𝐷 to the nearest hundredth.


𝐴𝐵 and 𝐴𝐶 are two chords in the circle 𝑀 in two opposite sides of its center, where 𝑚𝐵𝐴𝐶=33. If 𝐷 and 𝐸 are the midpoints of 𝐴𝐵 and 𝐴𝐶 respectively, find 𝑚𝐷𝑀𝐸.


Which of the choices below can be a chord length in a circle whose diameter is 19 cm?

  • A21 cm
  • B23 cm
  • C38 cm
  • D10 cm


A circle with center 𝑀 has a radius of 22 cm. The point 𝐴 lies 19 cm from 𝑀 and belongs to the chord 𝐵𝐶. Given that 𝐴𝐵=5𝐴𝐶, calculate the perpendicular distance between 𝑀 and the chord, giving your answer to the nearest whole number.

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