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 given figure, 𝐴𝐵 is a tangent to a circle with center 𝑀 at 𝐵, 𝐶𝐷 is a diameter, 𝑚𝐵𝐴𝑀=𝑥, and 𝑚𝑀𝐷𝐵=2𝑥55. Find the value of 𝑥 in degrees.

Q2:

In the given figure, 𝑃𝑄 is tangent to the circle with center 𝐵.

If the length of 𝐴𝑄 is 3, the length of 𝐵𝑃 is 5, and the length of 𝐴𝐵 is 40, what are the lengths of 𝐴𝑂 and 𝑂𝐵?

  • A30, 10
  • B20, 20
  • C15, 25
  • D10, 30

Q3:

The radii of the two circles on the common center 𝑀 are 3 cm and 4 cm. What is the length of 𝐴𝐵?

Q4:

Given that 𝑚𝑍𝑌𝐿=122, find 𝑚𝑋.

Q5:

Given that 𝐵𝐶 is a tangent to the circle with center 𝑀 and 𝑚𝐴𝑀𝐷=97, find 𝑚𝐶𝐵𝐷.

Q6:

Identify all the radii of circle 𝑀.

  • A 𝑀 𝐸 , 𝑀 𝐹 , 𝑀 𝐺
  • B 𝐸 𝐹 , 𝑀 𝐹
  • C 𝐹 𝐺

Q7:

If 𝐸𝐷 is a tangent to the circle at the point 𝐵, and 𝑚𝐸𝐵𝐶=40, what is 𝑚𝐴𝐶𝐵?

Q8:

Given that 𝐴𝐵 is a tangent to the circle 𝑀 at the point 𝐵, 𝑚𝐵𝑀𝐶=3𝑚𝐴, and the point 𝐶 is the midpoint of 𝐷𝐸, find the value of 𝑥.

Q9:

Given that 𝐴𝐷 is a tangent to the circle and 𝑚𝐷𝐴𝐶=90, calculate 𝑚𝐴𝐶𝐵.

Q10:

The circumference of a circle with center 𝑀 is 259 and 𝑍𝑌 is tangent at 𝑌. Calculate the length of 𝑍𝑌 to the nearest hundredth.

Q11:

Given that 𝑚𝐶𝐷𝐸=89, find 𝑚𝐷𝐵𝐶 and 𝑚𝐴𝐷𝐵.

  • A 7 6 , 3 8
  • B 5 1 , 8 9
  • C 8 9 , 5 1
  • D 7 6 , 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 𝐴𝐵 is a common tangent. Suppose 𝐴𝐵=𝑀𝑁=45.5cm and 𝐵𝐶=30.5cm. Find 𝐴𝑁 to the nearest tenth.

Q13:

Describe the position of the straight line 𝐿5𝑥+4𝑦1=0 with respect to the circle 𝑥+𝑦+6𝑥8𝑦+9=0.

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

Q14:

Where is the point (4,2) in relation to the circle (𝑥4)+(𝑦1)=1?

  • Aon
  • Binside
  • Coutside

Q15:

On which of the following does the point (3,6) lie?

  • Athe circle𝑥+𝑦=225
  • Bthe𝑦-axis
  • Cthe straight line 𝑦=2𝑥
  • Dthe𝑥-axis

Q16:

In the given figure, 𝐴𝐵 is a tangent to a circle with center 𝑀 at 𝐵, 𝐶𝐷 is a diameter, 𝑚𝐵𝐴𝑀=𝑥, and 𝑚𝑀𝐷𝐵=2𝑥5. Find the value of 𝑥 in degrees.

Q17:

Draw a triangle 𝐴𝐵𝐶 in which 𝐴𝐵=9cm, 𝐵𝐶=8cm, and 𝐶𝐴=7cm, and then draw a circle whose center is 𝐵 and radius is 5 cm. Determine whether point 𝐴 is located on, inside, or outside the circle.

  • AInside the circle
  • BOn the circle
  • COutside the circle

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