Worksheet: Positions of Points, Straight Lines, and Circles with respect to Circles

In this worksheet, we will practice finding the positions of points, straight lines, and circles with respect to other circles.

Q1:

In the figure below, 𝐴 𝐵 is a tangent to the circle 𝑀 at 𝐵 , 𝐶 𝐷 is a diameter, 𝑚 𝐵 𝐴 𝑀 = 𝑥 , and 𝑚 𝑀 𝐷 𝐵 = 2 𝑥 5 5 . Find the value of 𝑥 in degrees.

Q2:

In the given figure, 𝑃 𝑄 is tangent to the circle with centre 𝐵 .

If the length of 𝐴 𝑄 is 3, the length of 𝐵 𝑃 is 5, and the length of 𝐴 𝐵 is 40, what are the lengths of 𝐴 𝑂 and 𝑂 𝐵 ?

  • A 20, 20
  • B 10, 30
  • C 30, 10
  • D 15, 25

Q3:

The radii of the two circles on the common centre 𝑀 are 3 cm and 4 cm. What is the length of 𝐴 𝐵 ?

Q4:

Given that 𝑚 𝑍 𝑌 𝐿 = 1 2 2 , find 𝑚 𝑋 .

Q5:

Given that 𝐵 𝐶 is a tangent to the circle with centre 𝑀 and 𝐴 𝑀 𝐷 = 9 7 , find 𝐶 𝐵 𝐷 .

Q6:

Identify all the radii of circle .

  • A
  • B
  • C

Q7:

If 𝐸 𝐷 is a tangent to the circle at the point 𝐵 , and 𝑚 𝐸 𝐵 𝐶 = 4 0 , what is 𝑚 𝐴 𝐶 𝐵 ?

Q8:

Given that 𝐴 𝐵 is a tangent to the circle 𝑀 at the point 𝐵 , 𝑚 𝐵 𝑀 𝐶 = 3 𝑚 𝐴 , and the point 𝐶 is the midpoint of 𝐷 𝐸 , find the value of 𝑥 .

Q9:

Given that 𝐴 𝐷 is a tangent to the circle and 𝑚 𝐷 𝐴 𝐶 = 9 0 , calculate 𝑚 𝐴 𝐶 𝐵 .

Q10:

The circumference of circle 𝑀 is 259 and 𝑍 𝑌 is tangent at 𝑌 . Calculate the length of 𝑍 𝑌 to the nearest hundredth.

Q11:

Given that 𝑚 𝐶 𝐷 𝐸 = 8 9 , find 𝑚 𝐷 𝐵 𝐶 and 𝑚 𝐴 𝐷 𝐵 .

  • A 7 6 , 5 1
  • B 5 1 , 8 9
  • C 7 6 , 3 8
  • D 8 9 , 5 1

Q12:

In the figure, two circles with centres 𝑀 and 𝑁 touch externally at 𝐴 which is a point on the common tangent 𝐿 , where 𝐴 𝐵 is a common tangent. Suppose 𝐴 𝐵 = 𝑀 𝑁 = 4 5 . 5 c m and 𝐵 𝐶 = 3 0 . 5 c m . Find 𝐴 𝑁 to the nearest tenth.

Q13:

Describe the position of the straight line 𝐿 5 𝑥 + 4 𝑦 1 = 0 with respect to the circle 𝑥 + 𝑦 + 6 𝑥 8 𝑦 + 9 = 0 2 2 .

  • AThe straight line is a tangent to the circle.
  • BThe straight line is outside the circle.
  • CThe straight line is a secant to the circle.

Q14:

Where is the point ( 4 , 2 ) in relation to the circle ( 𝑥 4 ) + ( 𝑦 1 ) = 1 2 2 ?

  • Aoutside
  • Binside
  • Con

Q15:

On which of the following does the point lie?

  • Athe -axis
  • Bthe -axis
  • Cthe circle
  • Dthe straight line

Q16:

In the figure below, 𝐴 𝐵 is a tangent to the circle 𝑀 at 𝐵 , 𝐶 𝐷 is a diameter, 𝑚 𝐵 𝐴 𝑀 = 𝑥 , and 𝑚 𝑀 𝐷 𝐵 = 2 𝑥 5 . Find the value of 𝑥 in degrees.

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