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.


A circle has a radius of 90 cm. A point lies on the circle at a distance of (3π‘₯βˆ’3) cm from the center. Which of the following is true?

  • Aπ‘₯=31
  • Bπ‘₯>31
  • Cπ‘₯<31


A line 𝐿 intersects a circle with center 𝑀. Point 𝐴 lies on 𝐿 and is inside the circle. If the radius of the circle is 8 cm, π‘€π΄βŸ‚πΏ, and we set 𝑀𝐴=(3π‘₯βˆ’5)cm, in what interval does the value of π‘₯ belong?

  • Aο€Όβˆ’53,133
  • Bο€Ό53,133
  • Cο”βˆ’53,133
  • D53,133


Consider a circle of center 𝑀 with a radius π‘Ÿ and a point 𝐴 on a line 𝐿, where the line 𝐿doesnotintersectthecircle and π‘€π΄βŸ‚πΏ. Which of the following is true?

  • A𝑀𝐴<π‘Ÿ
  • B𝑀𝐴=π‘Ÿ
  • C𝑀𝐴>π‘Ÿ


Circle 𝑀 has radius 65. Suppose 𝐴 is on a line 𝐿 and 𝑀𝐴 is perpendicular to 𝐿. If 2π‘€π΄βˆ’56=18 what can be said of how 𝐿 lies with respect to the circle?

  • A𝐿 is a tangent to circle𝑀
  • B𝐿 is a secant to circle𝑀
  • C𝐿 is outside of circle𝑀


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·.


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.


Consider two circles 𝑀 and 𝑁 with a common tangent 𝐿 at point 𝐴. How many elements are there in the set π‘€βˆ©π‘?


Suppose we have two circles, one with center π‘€οŠ§ and radius π‘Ÿ=7 and one with center π‘€οŠ¨ and radius π‘Ÿ=4. Given that the circles intersect at two distinct points, which of the following is the correct range of values for the length π‘€π‘€οŠ§οŠ¨?

  • A𝑀𝑀<3
  • B3<π‘€π‘€οŠ§οŠ¨
  • C𝑀𝑀<11
  • D4<𝑀𝑀<7
  • E3<𝑀𝑀<11


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‘.


Given that 𝑀𝐢=16cm and 𝑁𝐡=13cm, find the length of 𝐢𝐡.

  • A29 cm
  • B8√13 cm
  • C26 cm
  • D832 cm

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