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

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


In the given figure, 𝐴𝐡 is a tangent to a circle with center 𝑀 at 𝐡, 𝐢𝐷 is a diameter, π‘šβˆ π΅π΄π‘€=π‘₯, and π‘šβˆ π‘€π·π΅=2π‘₯βˆ’55. Find the value of π‘₯ in degrees.


Given that ⃖⃗𝐡𝐢 is a tangent to the circle with center 𝑀 and π‘šβˆ π΄π‘€π·=97∘, find π‘šβˆ πΆπ΅π·.


Given that 𝐴𝐡 is a tangent to the circle 𝑀 at the point 𝐡, π‘šβˆ π΅π‘€πΆ=3π‘šβˆ π΄, and the point 𝐢 is the midpoint of 𝐷𝐸, find the value of π‘₯.


The circumference of a circle with center 𝑀 is 259 and οƒ«π‘π‘Œ is tangent at π‘Œ. Calculate the length of π‘π‘Œ to the nearest hundredth.


In the figure, two circles with centres 𝑀 and 𝑁 touch externally at 𝐴 which is a point on the common tangent ⃖⃗𝐿, where 𝐴𝐡 is a common tangent. Suppose 𝐴𝐡=𝑀𝑁=45.5cm and 𝐡𝐢=30.5cm. Find 𝐴𝑁 to the nearest tenth.


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

  • Athe circleΒ π‘₯+𝑦=225
  • Bthe 𝑦-axis
  • Cthe straight line 𝑦=2π‘₯
  • DtheΒ π‘₯-axis


Draw a triangle 𝐴𝐡𝐢 in which 𝐴𝐡=9cm, 𝐡𝐢=8cm, and 𝐢𝐴=7cm, and then draw a circle whose center is 𝐡 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


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


Circles 𝑀 and 𝑁 have radii of 29 cm and 18 cm and touch internally at 𝐴. Given that ⃖⃗𝐴𝐡 is a common tangent to the circles and that △𝐡𝑀𝑁 has an area of 55 cm2, what is the length of 𝐴𝐡?


Two circles with centers 𝑀 and 𝑁 are tangent at 𝐴. Given that π‘šβˆ π΄π΅π‘€=58∘, find π‘šβˆ π΅π‘€π‘.

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