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Worksheet: Properties of Tangents

In this worksheet, we will practice using the properties of tangents of circles to find missing values.

Q1:

Circle 𝑀 has radius 11 cm. If 𝐢 𝐴 = 1 6 . 3 c m , what is 𝐴 𝐡 ? Answer to the nearest tenth.

Q2:

Given that is a tangent to the circle at , and , find .

  • A
  • B
  • C
  • D

Q3:

Find the value of π‘₯ .

Q4:

In the following figure, the line that appears to be tangent is tangent. Find π‘Ž , and then find 𝑏 to the nearest tenth.

  • A π‘Ž = 3 . 0 c m , 𝑏 = 4 . 8 c m
  • B π‘Ž = 1 . 5 c m , 𝑏 = 1 . 4 c m
  • C π‘Ž = 3 . 4 c m , 𝑏 = 2 . 3 c m
  • D π‘Ž = 1 . 5 c m , 𝑏 = 2 . 3 c m
  • E π‘Ž = 4 . 5 c m , 𝑏 = 6 . 8 c m

Q5:

What is the name of a line that meets a circle in exactly one point?

  • Aradius
  • Bchord
  • Carc
  • Dtangent
  • Esegment

Q6:

In the figure, the circle 𝑀 has radius 5 cm. Given that  𝐴 𝐷 is tangent at 𝐷 , 𝐴 𝐡 = 2 c m , and 𝐴 𝐢 = 8 c m , determine the length of 𝐴 𝐷 .

  • A √ 6 6 cm
  • B 16 cm
  • C √ 3 9 cm
  • D 4 cm

Q7:

A circle with center 𝑃 has a radius of 20 cm, and 𝐸 𝐷 is tangent to the circle at 𝐷 . A point 𝐹 lies on both the circle and the line segment 𝐸 𝑃 . If 𝐸 𝐷 = 2 1 c m , find 𝐸 𝐹 .

Q8:

Given that  𝐴 𝐡 and  𝐴 𝐢 are two tangents, find the length of 𝐡 𝐢 .

Q9:

Given that 𝐴 𝐢 = ( 2 π‘₯ βˆ’ 3 ) c m , find π‘₯ and 𝑦 to the nearest thousandth.

  • A π‘₯ = 1 . 5 0 0 , 𝑦 = 5 . 0 0 0
  • B π‘₯ = 2 4 . 0 0 0 , 𝑦 = 1 1 . 0 0 0
  • C π‘₯ = 1 9 . 0 0 0 , 𝑦 = 1 9 . 0 0 0
  • D π‘₯ = 1 1 . 0 0 0 , 𝑦 = 2 4 . 0 0 0

Q10:

Given that 𝐴 𝐡 and 𝐴 𝐢 are two tangent segments to a circle at the points 𝐡 and 𝐢 , and 𝐴 𝐡 = 9 1 c m , determine the length of 𝐴 𝐢 .

Q11:

βƒ–     βƒ— 𝐴 𝐡 and βƒ–     βƒ— 𝐴 𝐢 are tangents to circle 𝑀 at 𝐡 and 𝐢 . If the radius of the circle is 6 cm and π‘š ∠ 𝐡 𝐴 𝐢 = 6 0 ∘ , find the lengths of 𝑀 𝐴 and 𝐴 𝐡 .

  • A 𝑀 𝐴 = 6 √ 3 cm, 𝐴 𝐡 = 12 cm
  • B 𝑀 𝐴 = 1 2 √ 3 cm, 𝐴 𝐡 = 6 √ 3 cm
  • C 𝑀 𝐴 = 6 √ 3 cm, 𝐴 𝐡 = 1 2 √ 3 cm
  • D 𝑀 𝐴 = 12 cm, 𝐴 𝐡 = 6 √ 3 cm

Q12:

The two circles 𝑀 and 𝑁 are touching externally. βƒ–     βƒ— 𝐹 𝐴 is a common tangent to them at 𝐴 and 𝐡 respectively, βƒ–     βƒ— 𝐹 𝐢 is a common tangent to them at 𝐢 and 𝐷 respectively. Given that 𝐴 𝐡 = 1 1 . 0 1 c m , and 𝐢 𝐷 = ( 𝑦 βˆ’ 1 1 . 0 1 ) c m , find π‘₯ and 𝑦 .

  • A π‘₯ = 1 1 . 0 1 , 𝑦 = 1 2 . 3 1
  • B π‘₯ = 1 2 . 3 1 , 𝑦 = 1 1 . 0 1
  • C π‘₯ = 1 3 . 0 1 , 𝑦 = 1 7 . 3 1
  • D π‘₯ = 1 4 . 3 1 , 𝑦 = 1 6 . 0 1

Q13:

Given that 𝐹 𝐷 = 6 . 4 2 c m , determine the lengths of 𝐹 𝐢 and 𝐸 𝐡 .

  • A 𝐹 𝐢 = 3 1 . 3 9 c m , 𝐸 𝐡 = 3 1 . 3 9 c m
  • B 𝐹 𝐢 = 6 . 4 2 c m , 𝐸 𝐡 = 6 . 4 2 c m
  • C 𝐹 𝐢 = 7 . 1 6 c m , 𝐸 𝐡 = 6 . 4 2 c m
  • D 𝐹 𝐢 = 6 . 4 2 c m , 𝐸 𝐡 = 7 . 1 6 c m

Q14:

In the figure,  𝐴 𝐡 is tangent to circle 𝑀 at 𝐴 , 𝑀 𝐴 = 3 6 . 3 c m , π‘š ∠ 𝐴 𝐡 𝑀 = 3 0 ∘ , and 𝐴 𝐢 βŸ‚ 𝑀 𝐡 . Find 𝐴 𝐢 to the nearest tenth.

Q15:

Lines and are tangents to circle . If , what is ?

  • A
  • B
  • C
  • D

Q16:

Find the length of 𝐡 𝐢 .

Q17:

Given that and are two tangents to the circle at the points and , respectively, determine .

  • A
  • B
  • C
  • D

Q18:

In the figure, βƒ–      βƒ— 𝑋 π‘Œ is tangent to circle 𝑀 at 𝑋 , 𝑀 π‘Œ meets the circle at 𝑍 , 𝑋 π‘Œ = 2 1 c m , and π‘Œ 𝑍 = 1 1 c m . Find the circle’s diameter to the nearest tenth.

Q19:

If I draw two tangent segments from a point outside the circle, are they equal in length?

  • Ayes
  • Bno

Q20:

If I draw two tangents to a circle, one from each end of a diameter, are the tangents parallel?

  • Ayes
  • Bno

Q21:

Line βƒ–     βƒ— 𝐴 𝐢 is tangent to circle 𝑀 at 𝐴 . Given that 𝐡 𝑀 = 5 5 c m , 𝐴 𝐢 = 9 6 c m , what is 𝐡 𝐢 ?

Q22:

Line βƒ–     βƒ— 𝐴 𝐢 is tangent to circle 𝑀 at 𝐴 . Given that 𝐡 𝑀 = 1 2 c m , 𝐴 𝐢 = 4 5 c m , what is 𝐡 𝐢 ?

Q23:

In the figure, line 𝑋 π‘Œ is tangent to the circle at 𝑋 . What is the length 𝑍 π‘Œ ?

Q24:

If two circle intersect at two points, how many common tangents do they have?

Q25:

If two circles lie outside each other, how many common tangents do they have?

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