Chapter 5.5, Problem 31E

### 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 Exercises 27 to 33, give both exact solutions and approximate solutions to two decimal places. Given: Right ∆ A B C with m ∠ C = 90 ° and m ∠ B A C = 60 ° ; point D on B C ¯ ; A D ⃑ bisects ∠ B A C and A B = 12 Find: B D

To determine

To find:

BD in a right ABC such that mC=90° and mBAC=60°; point D on BC¯;AD

Bisects BAC and AB=12

Explanation

Approach:

For a right triangle, for which the measure of the interior angles 30Â°, 60Â°, and 90Â°; if â€˜aâ€™ is the length of measure of the shorter leg; opposite to the angle 30Â°, then the length of the other two sides is given by

Length of the longer leg (opposite to 60Â°) =a3

Length of the hypotenuse (opposite to 90Â°)=2a.

In general

Length of the longer leg =3Ã— (Length of the shorter leg)

Length of the hypotenuse =2Ã— (Length of the shorter leg)

Calculation:

Given,

A right âˆ†ABC

mâˆ C=90Â° and mâˆ BAC=60Â°; point D on BCÂ¯;ADâƒ‘

Bisects âˆ BAC and AB=12.

Since, ADâƒ‘ bisects âˆ A of the âˆ†ABC, we have

=12(60Â°)

First consider the right âˆ†ABC to find AC and BC.

The above âˆ† should be of the type 30Â°-60Â°-90Â° as such mâˆ A=60Â°, mâˆ C=90Â°.

Thus,

mâˆ B=30Â°

30Â°-60Â°-90Â° theorem.

In a right triangle whose angle measure 30Â°, 60Â°, and 90Â°, the hypotenuse has a length equal to twice the length of the shorter leg, and the longer leg is the product of 3 and the length of the shorter leg.

AB= Hypotenuse =12 units.

AC= Shorter leg, (opposite to 30Â°)

BC= Longer leg

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