Chapter 5.6, Problem 36E

### 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 , m ∠ B = 30 ∘ . Also, A B = 12 3 and A D → bisects ∠ B A C . Find B D .

To determine

To find:

BD.

Explanation

Given:

In a right Î”ABC with right âˆ C, ADâ†’ bisects âˆ BAC and mâˆ B=30âˆ˜,AB=123.

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 mâˆ B=30âˆ˜,AB=123.

From the figure, sin30âˆ˜=ACAB

12=AC1231232=AC63=AC

Given that ADâ†’ bisects âˆ BAC.

By angle bisector theorem, we get

ABAC=BDDC

â‡’ACDC=ABBD

We have AB=123,AC=63.

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