Lesson Worksheet: Special Segments in a Circle Mathematics

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.

  • A𝐸𝐵=7cm, 𝐸𝐴=8cm
  • B𝐸𝐵=6.12cm, 𝐸𝐴=9.14cm
  • C𝐸𝐵=8cm, 𝐸𝐴=7cm
  • D𝐸𝐵=9.14cm, 𝐸𝐴=6.12cm

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.

  • A𝐸𝐵=2cm, 𝐸𝐴=9cm
  • B𝐸𝐵=1cm, 𝐸𝐴=18cm
  • C𝐸𝐵=4.5cm, 𝐸𝐴=13.5cm
  • D𝐸𝐵=9cm, 𝐸𝐴=2cm
  • E𝐸𝐵=18cm, 𝐸𝐴=1cm

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:

𝐴𝐵 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.

  • A𝑁𝐵=86.5, 𝑁𝐶=86.5
  • B𝑁𝐵=26, 𝑁𝐶=86.5
  • C𝑁𝐵=26, 𝑁𝐶=82.5
  • D𝑁𝐵=86.5, 𝑁𝐶=82.5

Q13:

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

  • A𝐸𝐵=10cm, 𝐵𝐴=6cm
  • B𝐸𝐵=6cm, 𝐵𝐴=10cm
  • C𝐸𝐵=24cm, 𝐵𝐴=20cm
  • D𝐸𝐵=6cm, 𝐵𝐴=4cm
  • E𝐸𝐵=4cm, 𝐵𝐴=6cm

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.

  • A𝐶𝐷=2.45cm, 𝑥=12.94cm
  • B𝐶𝐷=15.10cm, 𝑥=10.58cm
  • C𝐶𝐷=114.00cm, 𝑥=113.26cm
  • D𝐶𝐷=7.55cm, 𝑥=13.00cm

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.

  • A19.6 cm, 20.2 cm
  • B3.4 cm, 26.6 cm
  • C3.4 cm, 20.5 cm
  • D11.5 cm, 20.2 cm

Q17:

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

Q18:

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

  • ANo
  • BYes

Q19:

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

  • A𝑥=2
  • B𝑥=0.55
  • C𝑥=46.2
  • D𝑥=1.05

Q20:

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

  • A14 cm
  • B30 cm
  • C5√14 cm
  • D25 cm

Q21:

𝐴𝐵 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

Q22:

𝐴𝐵 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

Q23:

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

Q24:

Find the value of 𝑥.

Q25:

Consider a triangle 𝐴𝐷𝐸 where 𝐸𝐷=41. The point 𝐶 satisfies 𝐶∈𝐸𝐷, 𝐶∉𝐸𝐷, and 𝐶𝐷=45. ⃖⃗𝐸𝐴 is a tangent to the circle that passes through 𝐴, 𝐷, and 𝐶. Find the length of 𝐸𝐴 to the nearest hundredth.

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