   Chapter 5.6, Problem 41E 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 the figure, the angle bisectors of Δ A B C intersect at a point in the interior of the triangle. If B C = 5 , B A = 6 , and C A = 4 , find:a) C D and D B (HINT: Use Theorem 5.6.3.)b) C E and E A c) B F and F A d) Use results from parts (a), (b), and (c) to show that B D D C ⋅ C E E A ⋅ A F F B = 1. To determine

a)

To find:

CD and DB.

Explanation

Given:

In the figure, the angle bisectors of ΔABC intersect at a point in the interior of the triangle. Also BC=5,BA=6,CA=4.

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:

From the figure, we see that AD¯ is the bisector of CAB.

By angle bisector theorem, we get

CDAC=DBAB

From the figure, DB=CBCD

To determine

b)

To find:

CE and EA.

To determine

c)

To find:

BF and FA.

To determine

d)

To prove:

BDDCCEEAAFFB=1.

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