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Worksheet: Relations between Arcs, Chords, and Diameters

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 𝑀 𝐡
  • B 𝐢 𝑀
  • C 𝐴 𝐷

Q2:

In the figure, circles 𝐽 and 𝐾 are congruent, 𝐴 𝐡 β‰… 𝐢 𝐷 , 𝐴 𝐡 = ( 3 π‘₯ + 7 ) c m , and 𝐢 𝐷 = ( 8 π‘₯ + 1 2 ) c m . Find the length of 𝐴 𝐡 .

Q3:

Points 𝑋 and π‘Œ are midpoints of segments 𝐴 𝐡 and 𝐢 𝐷 respectively. If 𝐴 𝐡 = 6 0 , what is 𝐢 π‘Œ ?

Q4:

The radius of the circle 𝑂 below is 87 cm, 𝐴 𝐡 βˆ₯ 𝐢 𝐷 , and 𝑋 is the midpoint of 𝐴 𝐡 . Given that  𝑋 𝑂 intersects 𝐢 𝐷 at π‘Œ and 𝑂 π‘Œ = 6 0 c m , find the length of π‘Œ 𝐢 .

Q5:

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

Q6:

Given 𝐴 𝑀 = 2 0 0 c m and 𝑀 𝐢 = 1 2 0 c m , find 𝐴 𝐡 .

Q7:

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

  • A22.00, 1.60
  • B5.71, 11.00
  • C11.43, 11.00
  • D11.43, 1.60

Q8:

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

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

Q9:

In the following figure, determine all the chord(s) of the circle.

  • A
  • B
  • C

Q10:

The distance between the two lines and is 13 and . If the radius of the circle is 10.5, what is the length of ? If necessary, round your answer to 2 decimal places.

  • A23.5
  • B24.7
  • C22
  • D16.49
  • E33.92

Q11:

Given that , and cm, determine the length of .

Q12:

Given that 𝐴 𝐡 = 𝐢 𝐷 , 𝐴 𝐡 = 1 5 c m , 𝑀 𝐹 = 4 π‘₯ c m , and 𝐢 𝐷 = ( 1 1 π‘₯ + 4 ) c m , determine the value of π‘₯ and the length of 𝐴 𝑀 .

  • A π‘₯ = 1 , 𝐴 𝑀 = 7 . 5 c m
  • B π‘₯ = 4 , 𝐴 𝑀 = 7 2 . 2 5 c m
  • C π‘₯ = 4 , 𝐴 𝑀 = 1 5 c m
  • D π‘₯ = 1 , 𝐴 𝑀 = 8 . 5 c m

Q13:

Given that 𝐴 𝐡 = 𝐢 𝐷 = ( 6 π‘₯ + 3 ) c m , 𝑀 𝐸 = ( 3 π‘₯ + 1 ) c m , and 𝑀 𝑂 = 4 c m , find the length of 𝐢 𝐷 .

Q14:

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

Q15:

In circle 𝑆 , π‘š 𝑃 𝑄 𝑅 = 6 2 ∘ . Find π‘š 𝑃 𝑄 .

Q16:

If π‘š ∠ 𝐡 𝐴 𝐢 = 2 5 ∘ , what is π‘š ∠ 𝐹 𝐷 𝐸 ?

  • A 1 0 0 ∘
  • B 5 0 ∘
  • C 2 5 ∘

Q17:

The radius of circle 𝑀 is 60.9 cm, and 𝐴 𝐡 = 8 4 c m . What is the area of β–³ 𝐴 𝐷 𝐡 ?

Q18:

In the figure, the two circles are concentric at 𝑀 and 𝐴 𝐡 = 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 2 lies on both of the smaller circles and is the centre of the larger circle.

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

  • A18.85
  • B28.27
  • C94.25
  • D56.55
  • E62.83

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 𝐡 𝐢 to the nearest tenth.

Q22:

In the figure, the smaller of the concentric circles is tangent to chord 𝐴 𝐡 at 𝐢 . If the large circle has radius 67 cm, and 𝐴 𝐡 = 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 π‘₯ that satisfies the data represented.

  • A ] 2 0 , 2 9 [
  • B ] 2 9 , 6 2 [
  • C ] 4 , 2 4 [
  • D ] 2 0 , 6 2 [

Q24:

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

Q25:

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

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

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