TANGENT LINES TO CIRCLES WORKSHEET

Problem 1 :

Determine if the line segment AB is tangent to circle P.  

Problem 2 :

Determine if the line segment YX is tangent to circle P.  

Problem 3 :

If the line segment JK is tangent to circle L, find x.   

Problem 4 :

If the line segment JK is tangent to circle L, find x.   

Problem 5 :

If the line segment JK is tangent to circle L, find x.   

Problem 6 :

If the line segment JK is tangent to circle L, find x. 

Problem 7 :

If the line segment JK is tangent to circle L, find x.   

Problem 8 :

If the line segment JK is tangent to circle L, find x.   

Problem 9 :

If the line segment JK is tangent to circle L, find x.   

Problem 10 :

Find the value of x. Assume that segment that appears to be tangent is tangent.   

Answers

1. Answer :

If the line segment AB is tangent to circle P, it is perpendicular to the radius PA.

Then, m∠PAB = 90° and triangle PAB has to be a right triangle. 

Using Pythagorean Theorem, verify whether triangle PAB is a right triangle.  

PA² + PB²  =  PB²

8² + 15²  =  17²  ?

64 + 225  =  289  ?

289  =  289  ?

The above result is true.

So, triangle PAB is a right triangle and m∠PAB = 90°.

Hence, the line segment AB is tangent to circle P.  

2. Answer :

Find the length of ZY : 

ZY  =  Radius + 5

ZY  =  8 + 5

ZY  =  13

If the line segment YX is tangent to circle Z, it is perpendicular to the radius ZX.

Then, m∠ZXY = 90° and triangle ZXY has to be a right triangle. 

Using Pythagorean Theorem, verify whether triangle ZXY is a right triangle.  

ZX² + XY²  =  ZY²

8² + 10²  =  13²  ?

64 + 100  =  169  ?

164  =  169  ?

The above result is false.

So, triangle ZXY is not a right triangle and m∠ZXY ≠ 90°.

Hence, the line segment YX is not tangent to circle Z.  

3. Answer :

Because JK is tangent to circle L, m∠LJK = 90° and triangle LJK is a right triangle. 

BY Pythagorean Theorem, 

LJ² + JK²  =  LK²

7² + 19²  =  x²

49 + 361  =  x²

49 + 361  =  x²

400  =  x²

Take square root on both sides. 

20  =  x

4. Answer :

Find the length of LK : 

LK  =  Radius + 7

LK  =  11 + 7

LK  =  18

Because JK is tangent to circle L, m∠LJK = 90° and triangle LJK is a right triangle. 

BY Pythagorean Theorem,

LJ² + JK²  =  LK²

11² + x²  =  18²

121 + x²  =  324

Subtract 121 from each side.

x²  =  203

Take square root on both sides. 

x  ≈  14.2

5. Answer :

Find the length of LK : 

LK  =  Radius + x

LK  =  10 + x

Because JK is tangent to circle L, m∠LJK = 90° and triangle LJK is a right triangle. 

BY Pythagorean Theorem,

LJ² + JK²  =  LK²

10² + 24²  =  (10 + x)²

676  =  (10 + x)²

Take square root on both sides. 

26  =  10 + x

Subtract 10 from each side.

16  =  x

6. Answer :

Find the length of JM : 

JM  =  JL + LM

JM  =  11 + 11

JM  =  22

Because JK is tangent to circle L, m∠MJK = 90° and triangle MJK is a right triangle. 

BY Pythagorean Theorem,

JM² + JK²  =  MK²

22² + x²  =  25²

484 + x²  =  625

Subtract 484 from each side.

x²  =  141

Take square root on both sides. 

x  ≈  11.9

7. Answer :

Find the length of JM : 

JM  =  JL + LM

JM  =  3.5 + 3.5

JM  =  7

Because JK is tangent to circle L, m∠MJK = 90° and triangle MJK is a right triangle. 

BY Pythagorean Theorem,

JM² + JK²  =  MK²

7² + 6.4²  =  x²

49 + 40.96  =  x²

89.96  =  x²

Take square root on both sides. 

9.5  ≈  x

8. Answer :

Find the length of JM : 

JM  =  JL + LM

JM  =  x + x

JM  =  2x

Because JK is tangent to circle L, m∠MJK = 90° and triangle MJK is a right triangle. 

BY Pythagorean Theorem,

JK² + JM²  =  KM²

7² + (2x)²  =  25²

49 + 4x²  =  625

Subtract 49 from each side.

4x²  =  576

Divide each side by 4.

x²  =  144

Take square root on both sides. 

x  =  12

9. Answer :

Find the length of LK : 

LK  =  Radius + 10

LK  =  x + 10

Because JK is tangent to circle L, m∠LJK = 90° and triangle LJK is a right triangle. 

BY Pythagorean Theorem,

LJ² + JK²  =  LK²

x² + 20²  =  (x + 10)²

x² + 400  =  x² + 2(x)(10) + 10²

x² + 400  =  x² + 20x + 100

Subtract x² from each side.

400  =  20x + 100

Subtract 100 from each side.

300  =  20x

Divide each side by 20.

15  =  x

10. Answer :

In the diagram above, the length of the missing leg of the triangle is x. Because its the radius of the circle. 

Since the line segment that appears to be tangent is tangent, the triangle is a right triangle.

By Pythagorean Theorem,

x² + 12²  =  (x + 6)²

x² + 144  =  x² + 2(x)(6) + 6²

x² + 144  =  x² + 12x + 36

Subtract x² from each side.

144  =  12x + 36

Subtract 36 from each side.

108  =  12x

Divide each side by 12.

9  =  x

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