Chapter 5.5, Problem 33E

### 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: ∆ A B C with m ∠ A = 45 ° , m ∠ B = 30 ° , and B C = 12 Find: A B (HINT: Use altitude C D ¯ from C to A B ¯ as an auxiliary line)

To determine

To find:

AB of the ABC with mA=45°, mB=30° and BC=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)

Approach:

For a right triangle for which the measure of the interior angles 45Â°, 45Â°, and 90Â°, if â€˜aâ€™ is the length of measure of one of the leg; opposite to the angle 45Â°, then the length of the other two sides is given by

Length of the other leg =a

Length of the hypotenuse =a2

In general

Length of the legs are equal.

Length of the hypotenuse =2Ã— (Length of one of the legs)

Calculation:

Given,

âˆ†ABC with

mâˆ A=45Â°,

mâˆ B=30Â° and

BC=12.

Let us find the third angle âˆ C of the âˆ†ABC.

By the property that the sum of all the three interior angle of a triangle is 180Â°.

âˆ A+âˆ B+âˆ C=180Â°

45Â°+30Â°+âˆ C=180Â°

75Â°+âˆ C=180Â°

âˆ C=105Â°

Draw the altitude CDÂ¯ from C to ABÂ¯, which divides the âˆ†ABC into two right triangles âˆ†ACD and right âˆ†BCD and also divides the angle at the vertex C of the âˆ†ABC into two parts.

One part of the angle âˆ C in the right âˆ†ACD is 45Â° and the other part should be 60Â°, which lies in the right âˆ†BCD such that the sum of all the three interior angles in both the right âˆ†ACD and in the right âˆ†BCD become 180Â°

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