# 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.

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.

Find the length of ZY :

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.

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

Find the length of LK :

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

Find the length of LK :

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

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

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

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

Find the length of LK :

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

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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