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.

Q1:

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

Q2:

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

Q3:

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.

Q4:

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

Q5:

Given 𝐴𝑀=200cm and 𝑀𝐶=120cm, find 𝐴𝐵.

Q6:

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

Q7:

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

Q8:

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.

Q9:

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

Q10:

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

Q11:

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

Q12:

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

Q13:

In circle 𝑆, 𝑚𝑃𝑄𝑅=62. Find 𝑚𝑃𝑄.

Q14:

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

Q15:

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

Q16:

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

Q17:

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?

Q18:

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)

Q19:

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?

Q20:

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.

Q21:

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?

Q22:

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.

Q23:

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.

Q24:

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

Q25:

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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