   Chapter 5.6, Problem 36E Elementary Geometry For College St...

7th Edition
Alexander + 2 others
ISBN: 9781337614085

Solutions

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 ΔABC with right C, AD bisects BAC and mB=30,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 ΔABC.

Given that mB=30,AB=123.

From the figure, sin30=ACAB

12=AC1231232=AC63=AC

Given that AD bisects BAC.

By angle bisector theorem, we get

ABAC=BDDC

ACDC=ABBD

We have AB=123,AC=63.

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