Chapter 5.5, Problem 21E

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

Given: ∆ X Y Z with anglesas shown in thedrawing Find: X Y (HINT: Compare this drawing to the one for Exercise 20.)

To determine

To find:

XY in XYZ with angles as shown in the drawing.

Explanation

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,

In âˆ†XYZ,

mâˆ X=60Â°

mâˆ Y=45Â°, and

mâˆ Z=75Â°.

Also, ZX=12 units.

Now, draw a perpendicular (ZW) from the vertex Z to the base XY of the âˆ†XYZ, which divides the angle 75Â° at the vertex Z into two parts such that

mâˆ YZW=45Â° and mâˆ XZW=30Â°

Thus, âˆ†XYZ is divided into two right triangles as âˆ†YWZ and âˆ†WXZ.

Here, âˆ†YWZ is of 45Â°-45Â°-90Â° type of right and âˆ†MPQ is of 30Â°-60Â°-90Â° type of right triangle.

Now, consider the triangle âˆ†WXZ which has XZ is the hypotenuse and WZ is the leg opposite to the 60Â° angle and which is the longer leg and the leg WX is the shorter leg.

Thus, in âˆ†WXZ

XZ= Hypotenuse =12 units.

WX= Shorter leg, and

WZ= 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

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