Worksheet: Relations between Arcs, Chords, and Diameters

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In this worksheet, we will practice identifying arcs, chords, and diameters and using the relations between them to solve problems.

Q1:

Which segment in this circle is a chord?

  • A 𝐶 𝑀
  • B 𝐴 𝐷
  • C 𝑀 𝐵

Q2:

In the figure, circles 𝐽 and 𝐾 are congruent, 𝐴 𝐵 ≅ 𝐶 𝐷 , 𝐴 𝐵 = ( 3 𝑥 + 7 ) c m , and 𝐶 𝐷 = ( 8 𝑥 + 1 2 ) c m . Find the length of 𝐴 𝐵 .

Q3:

Points 𝑋 and 𝑌 are midpoints of segments 𝐴 𝐵 and 𝐶 𝐷 respectively. If 𝐴 𝐵 = 6 0 , what is 𝐶 𝑌 ?

Q4:

The radius of the circle 𝑂 below is 87 cm, 𝐴 𝐵 ∥ 𝐶 𝐷 , and 𝑋 is the midpoint of 𝐴 𝐵 . Given that 𝑋 𝑂 intersects 𝐶 𝐷 at 𝑌 and 𝑂 𝑌 = 6 0 c m , find the length of 𝑌 𝐶 .

Q5:

𝐴 𝐵 is a chord in circle 𝑀 whose radius is 25.5 cm. If 𝐴 𝐵 = 4 0 . 8 c m , what is the length of 𝐷 𝐸 ?

Q6:

Given 𝐴 𝑀 = 2 0 0 c m and 𝑀 𝐶 = 1 2 0 c m , find 𝐴 𝐵 .

Q7:

In circle 𝑀 , suppose 𝑀 𝐴 = 1 1 and 𝑀 𝐶 = 9 . 4 . Find the lengths of 𝐴 𝐵 and 𝐶 𝐷 to the nearest hundredth.

  • A11.43, 11.00
  • B22.00, 1.60
  • C11.43, 1.60
  • D5.71, 11.00

Q8:

In the given circle, 𝑂 𝐴 = 8 . 5 cm and 𝑂 𝐶 = 4 cm. Determine the lengths of 𝐴 𝐵 and 𝐶 𝐷 .

  • A 𝐴 𝐵 = 1 5 cm, 𝐶 𝐷 = 4 . 5 c m
  • B 𝐴 𝐵 = 7 . 5 c m , 𝐶 𝐷 = 4 . 5 c m
  • C 𝐴 𝐵 = 1 5 c m , 𝐶 𝐷 = 4 c m
  • D 𝐴 𝐵 = 7 . 5 c m , 𝐶 𝐷 = 4 c m

Q9:

In the following figure, determine all the chords of the circle.

  • A 𝑁 𝐶 , 𝑁 𝐵 , 𝑁 𝐸 , 𝑁 𝐹
  • B 𝐵 𝐸 , 𝐹 𝐶
  • C 𝐵 𝐶 , 𝐹 𝐷
  • D 𝐵 𝐸 , 𝐹 𝐶 , 𝐵 𝐶 , 𝐹 𝐷

Q10:

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

Q11:

Given that 𝑀 𝑋 = 𝑀 𝑌 , and 𝑌 𝐷 = 3 1 cm, determine the length of 𝐴 𝐵 .

Q12:

Given that 𝐴 𝐵 = 𝐶 𝐷 , 𝐴 𝐵 = 1 5 c m , 𝑀 𝐹 = 4 𝑥 c m , and 𝐶 𝐷 = ( 1 1 𝑥 + 4 ) c m , determine the value of 𝑥 and the length of 𝐴 𝑀 .

  • A 𝑥 = 1 , 𝐴 𝑀 = 7 . 5 c m
  • B 𝑥 = 4 , 𝐴 𝑀 = 1 5 c m
  • C 𝑥 = 4 , 𝐴 𝑀 = 7 2 . 2 5 c m
  • D 𝑥 = 1 , 𝐴 𝑀 = 8 . 5 c m

Q13:

Given that 𝐴 𝐵 = 𝐶 𝐷 = ( 6 𝑥 + 3 ) c m , 𝑀 𝐸 = ( 3 𝑥 + 1 ) c m , and 𝑀 𝑂 = 4 c m , find the length of 𝐶 𝐷 .

Q14:

In the circle 𝑀 , if 𝐴 𝐵 = 1 8 . 4 c m , find the length of 𝐶 𝐵 .

Q15:

In circle 𝑆 , 𝑚 𝑃 𝑄 𝑅 = 6 2 ∘ . Find 𝑚 𝑃 𝑄 .

Q16:

If 𝑚 ∠ 𝐵 𝐴 𝐶 = 2 5 ∘ , what is 𝑚 ∠ 𝐹 𝐷 𝐸 ?

Q17:

The radius of circle 𝑀 is 60.9 cm, and 𝐴 𝐵 = 8 4 c m . What is the area of △ 𝐴 𝐷 𝐵 ?

Q18:

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

Q19:

Two circles have been cut out of the bigger circle as seen in the picture. The point 𝑂 lies on both of the smaller circles and is the center of the larger circle.

Work out the remaining area. Give your answer accurate to two decimal places.

Q20:

Two small circles with the same diameter are drawn inside a larger circle as shown. Given that the diameter of the large circle is 9.8 mm, determine, to the nearest tenth, the area of the shaded part.

Q21:

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

Q22:

In the figure, the smaller of the concentric circles is tangent to chord 𝐴 𝐵 at 𝐶 . If the large circle has radius 67 cm, and 𝐴 𝐵 = 1 2 0 c m , what is the radius of the smaller circle?

Q23:

If 𝑀 𝐹 > 𝑀 𝐸 , find the range of values of 𝑥 that satisfies the data represented.

  • A ( 4 , 2 4 )
  • B ( 2 0 , 6 2 )
  • C ( 2 9 , 6 2 )
  • D ( 2 0 , 2 9 )

Q24:

Point 𝑀 is at a distance 13 from 𝑂 , which is the center of a circle of radius 7.3. Draw ray 𝑀 𝐵 to intersect the circle at 𝐴 and 𝐵 , with 𝐴 between 𝑀 and 𝐵 . If 𝑀 𝐴 = 5 . 7 , what is 𝐴 𝐵 ?

Q25:

Given that the circle 𝑀 shown below has a radius of 25 cm, 𝐴 𝐵 = 3 6 c m , and 𝐶 𝐷 = 4 8 c m , find the length of 𝑀 𝑂 .

  • A 25 cm
  • B 5 √ 1 4 cm
  • C 14 cm
  • D 30 cm

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