Chapter 5.6, Problem 38E

### Elementary Geometry For College St...

7th Edition
Alexander + 2 others
ISBN: 9781337614085

Chapter
Section

### Elementary Geometry For College St...

7th Edition
Alexander + 2 others
ISBN: 9781337614085
Textbook Problem

# In right Δ A B C with right ∠ C , A D → bisects ∠ B A C . If A C = 6 and D C = 3 , find B D and A B . (HINT: Let B D = x and A B = 2 x . Then use the Pythagorean Theorem.)

To determine

To find:

BD and AB.

Explanation

Given:

In a right Î”ABC with right âˆ C, ADâ†’ bisects âˆ BAC and AC=6,DC=3.

Theorem used:

Angle bisector theorem:

If a ray bisects one angle of a triangle, then it divides the opposite side into segments whose lengths are proportional to the lengths of the two sides that form the bisected angle.

Calculation:

Consider a right Î”ABC.

Given that ADâ†’ bisects âˆ BAC.

By angle bisector theorem, we get

ABAC=BDDC

â‡’ACDC=ABBD

We have AC=6,DC=3.

63=ABBD21=ABBD

â‡’AB:BD=2:1

Let AB=2x,BD=x

From the figure we see that, BC=BD+DC

â‡’BC=x+3

Consider the right triangle ABC with right angle at C

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