Lesson Worksheet: Perpendicular Bisector of a Chord Mathematics

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.

Question 1

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

Question 2

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

Question 3

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

Question 4

Given 𝐴𝑀=200cm and 𝑀𝐶=120cm, find 𝐴𝐵.

Question 5

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

Question 6

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

Question 7

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.

Question 8

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

Question 9

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

Question 10

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

Question 11

In circle 𝑆, 𝑚𝑃𝑄𝑅=62∘. Find 𝑚𝑃𝑄.

Question 12

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

Question 13

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

Question 14

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

Question 15

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?

Question 16

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?

Question 17

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.

Question 18

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

Question 19

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.

Question 20

𝐴𝐵 and 𝐴𝐶 are two chords in the circle 𝑀 in two opposite sides of its center, where 𝑚∠𝐵𝐴𝐶=33∘. If 𝐷 and 𝐸 are the midpoints of 𝐴𝐵 and 𝐴𝐶 respectively, find 𝑚∠𝐷𝑀𝐸.

Question 21

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

Question 22

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.

Question 23

A circle of center 𝑀 has a radius of 13 and a point 𝐴 is at a distance of 20 from its center. A line that is a tangent to the circle at a point 𝐷 is drawn through 𝐴. Calculate the length of 𝐴𝐷. Round your answer to the nearest whole number.

Question 24

In a circle of center 𝑂, 𝐴𝐵=35cm, 𝐶𝐵=25cm, and 𝐴𝐶=40cm. Given that 𝑂𝐷⟂𝐵𝐶 and 𝑂𝐸⊥𝐴𝐶, find the perimeter of △𝐶𝐷𝐸.

Question 25

Using the figure and the fact that 𝐵𝐶=32√3cm, determine the diameter of the circle.

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