Chapter 5.5, Problem 27E

### 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: In ∆ A B C , A D ⃑ bisects ∠ B A C m ∠ B = 30 ° and A B = 12 Find: D C and D B

To determine

To find:

DC and DB such that in ABCD, AD bisects BAC, mB=30° 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,

The âˆ†ABC with bisector of the angle âˆ BAC.

mâˆ B=30Â°

AB=12

Since, one of the acute angle mâˆ C of the right triangle ABC is 30Â°, then the ofter acute angle mâˆ A should be 60Â°

Thus, the âˆ†ABC is of the type 30Â°-60Â°-90Â° triangle.

Since, the interior angle âˆ A of the triangle âˆ†ABC is bisected by the ray ADâƒ‘, we have

Thus,

Now, consider the right triangle ABC

Thus, âˆ†ABC is of the type 30Â°-60Â°-90Â° triangle.

AB= Hypotenuse =12 units.

CA= Shorter leg,

BC= Longer leg.

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.

The Hypotenuse =2Ã— (Length of shorter leg)

AB=2Ã—(AC)

12=2Ã—(AC)

Divide; by 2 on both sides,

122=2Ã—AC2

AC=6 units

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