ANGLE BISECTOR THEOREM WORKSHEET

1. In the ΔABC shown below, find the length of BD.

2. In the ΔABC shown below, find the length of CD.

3. Solve for x.

4. Solve for x.

5. Solve for x.

6. Solve for x.

1. Answer :

Since AD is the angle bisector of ∠A, by Angle Bisector Theorem,

BD/DC = AB/AC

Substitute.

BD/6 = 4/8

BD/6 = 1/2

Multiply each side by 6.

BD = 3

2. Answer :

Let x be the length of CD.

Then the length of DB = 30 - x.

Since AD is the angle bisector of ∠A, by Angle Bisector Theorem,

CD/DB = AC/AB

Substitute.

x/(30 - x) = 12/28

x/(30 - x) = 3/7

7x = 3(30 - x)

7x = 90 - 3x

Add 3x to each side.

10x = 90

Divide each side by 10.

x = 9

CD = 9

3. Answer :

Since BD is the angle bisector of ∠B, by Angle Bisector Theorem,

CD/DA = BC/BA

Substitute.

(x + 1)/21 = 15/35

(x + 1)/21 = 3/7

7(x + 1) = 3(21)

7x + 7 = 63

Subtract 7 from each side.

7x = 56

Divide each side by 7.

x = 8

4. Answer :

Since CD is the angle bisector of ∠C, by Angle Bisector Theorem,

AD/DB = CA/CB

Substitute.

2/4 = 5/(2x - 2)

1/2 = 5/(2x - 2)

1(2x - 2) = 5(2)

2x - 2 = 10

Add 2 to each side.

2x = 12

Divide each side by 2.

x = 6

5. Answer :

Find the length of DB :

DB = AB - AD

= (x + 3) - 4

= x + 3 - 4

= x - 1

Since CD is the angle bisector of ∠C, by Angle Bisector Theorem,

AD/DB = CA/CB

Substitute.

4/(x - 1) = 6/9

4/(x - 1) = 2/3

3(4) = 2(x - 1)

12 = 2x - 2

Add 2 to each side.

14 = 2x

Divide each side by 2.

7 = x

6. Answer :

Find the length of AD :

AD = AC - DC

= 18 - 8

= 10

Since BD is the angle bisector of ∠B, by Angle Bisector Theorem,

CD/DA = BC/BA

Substitute.

8/10 = (2x - 4)/15

4/5 = (2x - 4)/15

15(4) = 5(2x - 4)

60 = 10x - 20

Add 20 to each side of the equation.

80 = 10x

Divide each side by 10.

8 = x

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