# Chapter 5.5, Problem 30E

### 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
To determine

To find:

MN of the isosceles right triangle STV with M and N are the midpoints of ST¯ and SV¯.

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,

âˆ†STV is an isosceles triangle with right angle at T.

M and N are the midpoints of STÂ¯ and SVÂ¯.

SV=20 units.

Since, STV is an isosceles right âˆ†, two of the sides must be congruent other than the hypotenuse.

From the given diagram SVÂ¯.is the hypotenuse.

Thus,

TSÂ¯â‰…TVÂ¯

Since, two of the legs of the right âˆ†STV are congruent, the angle opposite to those congruent sides must be congruent.

Hence, âˆ†STV is of the 45Â°-45Â°-90Â° type of right triangle.

By the application of the 45-45-90 theorm,

45Â°-45Â°-90Â° theorem.

In a right triangle whose angle measures 45Â°, 45Â°, and 90Â°, the legs are congruent and the hypotenuse has a length equal to the product of 2 and the length of either leg

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