Worksheet: Special Segments in a Circle

In this worksheet, we will practice using the theorems of intersecting chords, secants, or tangents and secants to find missing lengths in a circle.

Q1:

If 𝐸𝐴𝐸𝐵=87, 𝐸𝐶=7cm, and 𝐸𝐷=8cm, find the lengths of 𝐸𝐵 and 𝐸𝐴.

  • A 𝐸 𝐵 = 7 c m , 𝐸 𝐴 = 8 c m
  • B 𝐸 𝐵 = 6 . 1 2 c m , 𝐸 𝐴 = 9 . 1 4 c m
  • C 𝐸 𝐵 = 8 c m , 𝐸 𝐴 = 7 c m
  • D 𝐸 𝐵 = 9 . 1 4 c m , 𝐸 𝐴 = 6 . 1 2 c m

Q2:

Given that 𝐸𝐴=5.2cm, 𝐸𝐶=6cm, 𝐸𝐵=7.5cm, and 𝐸𝐷=6.5cm, do the points 𝐴, 𝐵, 𝐶, and 𝐷 lie on a circle?

  • Ayes
  • Bno

Q3:

In the following figure, find the value of 𝑥.

Q4:

Given that 𝐸𝐶=4, 𝐸𝐷=15, and 𝐸𝐵=6, find the length of 𝐸𝐴.

Q5:

Points 𝐴, 𝐸, 𝐵 and 𝐷, 𝐸, 𝐶 are collinear. Suppose that 𝐸𝐴=7.3, 𝐸𝐶=5.2, 𝐸𝐵=5.8, and 𝐸𝐷=7.6. Do points 𝐴, 𝐵, 𝐶, and 𝐷 lie on a circle?

  • Ano
  • Byes

Q6:

What is the radius of circle 𝑀?

Q7:

In the following figure, find the value of 𝑥.

Q8:

Given that 𝐸𝐶=12, 𝐸𝐷=11, and 𝐸𝐵=12, find the length of 𝐸𝐴.

Q9:

If 𝐸𝐴𝐸𝐵=92, 𝐸𝐶=3cm, and 𝐸𝐷=6cm, find the lengths of 𝐸𝐵 and 𝐸𝐴.

  • A 𝐸 𝐵 = 2 c m , 𝐸 𝐴 = 9 c m
  • B 𝐸 𝐵 = 1 c m , 𝐸 𝐴 = 1 8 c m
  • C 𝐸 𝐵 = 4 . 5 c m , 𝐸 𝐴 = 1 3 . 5 c m
  • D 𝐸 𝐵 = 9 c m , 𝐸 𝐴 = 2 c m
  • E 𝐸 𝐵 = 1 8 c m , 𝐸 𝐴 = 1 c m

Q10:

Given that 𝐸𝐴=7.5cm, 𝐸𝐶=5.6cm, 𝐸𝐵=7.9cm, and 𝐸𝐷=6cm, do the points 𝐴, 𝐵, 𝐶, and 𝐷 lie on a circle?

  • Ano
  • Byes

Q11:

Points 𝐴, 𝐸, 𝐵 and 𝐷, 𝐸, 𝐶 are collinear. Suppose that 𝐸𝐴=7.4, 𝐸𝐶=5.4, 𝐸𝐵=6, and 𝐸𝐷=7. Do points 𝐴, 𝐵, 𝐶, and 𝐷 lie on a circle?

  • Ano
  • Byes

Q12:

𝐴 𝐵 and 𝐴𝐶 are two chords of a circle of center 𝑀 and radius 26. Suppose that 𝑀𝐾 is perpendicular to both 𝐴𝐵 and 𝐴𝐶 and contains a point 𝑁 with 𝑀𝑁=82.5. Find the lengths of 𝑁𝐵 and 𝑁𝐶.

  • A 𝑁 𝐵 = 8 6 . 5 , 𝑁 𝐶 = 8 6 . 5
  • B 𝑁 𝐵 = 2 6 , 𝑁 𝐶 = 8 6 . 5
  • C 𝑁 𝐵 = 2 6 , 𝑁 𝐶 = 8 2 . 5
  • D 𝑁 𝐵 = 8 6 . 5 , 𝑁 𝐶 = 8 2 . 5

Q13:

If 𝐸𝐴𝐸𝐵=53, 𝐸𝐶=12cm, and 𝐸𝐷=5cm, find the lengths of 𝐸𝐵 and 𝐵𝐴.

  • A 𝐸 𝐵 = 1 0 c m , 𝐵 𝐴 = 6 c m
  • B 𝐸 𝐵 = 6 c m , 𝐵 𝐴 = 1 0 c m
  • C 𝐸 𝐵 = 2 4 c m , 𝐵 𝐴 = 2 0 c m
  • D 𝐸 𝐵 = 6 c m , 𝐵 𝐴 = 4 c m
  • E 𝐸 𝐵 = 4 c m , 𝐵 𝐴 = 6 c m

Q14:

A circle has center 𝑀 and radius 13 cm. A line passes through the points 𝐵, 𝐶, and 𝐷 where 𝐶 and 𝐷 are on the circle, 𝐵 is 25 cm from the point 𝑀, and 𝐶𝐵=𝐶𝐷. Calculate the length of 𝐶𝐷 and the perpendicular distance 𝑥 between the line and the point 𝑀. Round your answers to 2 decimal places.

  • A 𝐶 𝐷 = 2 . 4 5 c m , 𝑥 = 1 2 . 9 4 c m
  • B 𝐶 𝐷 = 1 5 . 1 0 c m , 𝑥 = 1 0 . 5 8 c m
  • C 𝐶 𝐷 = 1 1 4 . 0 0 c m , 𝑥 = 1 1 3 . 2 6 c m
  • D 𝐶 𝐷 = 7 . 5 5 c m , 𝑥 = 1 3 . 0 0 c m

Q15:

Given that the points 𝐴, 𝐵, 𝐶, and 𝐷 lie on a circle, find the length of 𝐵𝐴.

Q16:

In the figure shown, the circle has a radius of 12 cm, 𝐴𝐵=12cm, and 𝐴𝐶=35cm. Determine the distance from 𝐵𝐶 to the center of the circle, 𝑀, and the length of 𝐴𝐷, rounding your answers to the nearest tenth.

  • A 19.6 cm, 20.2 cm
  • B 3.4 cm, 26.6 cm
  • C 3.4 cm, 20.5 cm
  • D 11.5 cm, 20.2 cm

Q17:

In the figure below, 𝐵𝐶 is a diameter of the circle 𝑀, 𝐴𝐵=𝐴𝐶, 𝑚𝐵𝐴𝐶=60, 𝐵𝑋=22.9cm, and 𝑀𝑋𝐴𝐵. Find the length of 𝐴𝐸.

Q18:

If 𝐸𝐶=10cm, 𝐸𝐷=6cm, 𝐸𝐵=5cm, find the length of 𝐸𝐴.

Q19:

Are the points 𝐴, 𝐵, 𝐶, and 𝐷 lying on a circle?

  • Ano
  • Byes

Q20:

Given that 𝐸𝐴=11𝑥, 𝐸𝐵=21𝑥, 𝐸𝐶=22, and 𝐸𝐷=42, find the value of 𝑥.

  • A 𝑥 = 2
  • B 𝑥 = 0 . 5 5
  • C 𝑥 = 4 6 . 2
  • D 𝑥 = 1 . 0 5

Q21:

Given that the circle 𝑀 shown below has a radius of 25 cm, 𝐴𝐵=36cm, and 𝐶𝐷=48cm, find the length of 𝑀𝑂.

  • A14 cm
  • B30 cm
  • C 5 1 4 cm
  • D25 cm

Q22:

𝐴 𝐵 and 𝐶𝐷 are two chords of a circle intersecting at 𝐻. Find 𝐻𝐷, where 𝐴𝐻=10, 𝐻𝐵=8, and 𝐶𝐻=16.

  • A2
  • B20
  • C5
  • D12.8
  • E5.6

Q23:

𝐴 𝐵 and 𝐶𝐷 are two intersecting line segments at 𝐻 inside a circle, where 𝐴𝐻=4, 𝐻𝐵=6, 𝐷𝐻=3, and 𝐻𝐶=8. Are 𝐴𝐵 and 𝐶𝐷 chords?

  • AYes
  • BNo

Q24:

In the given figure, 𝐴𝐵=5cm, 𝐶𝐷=9cm, and 𝐷𝐸=3cm. Find the length of 𝐵𝐸.

Q25:

Find the value of 𝑥.

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