   Chapter 5.5, Problem 19E ### Elementary Geometry for College St...

6th Edition
Daniel C. Alexander + 1 other
ISBN: 9781285195698

#### Solutions

Chapter
Section ### Elementary Geometry for College St...

6th Edition
Daniel C. Alexander + 1 other
ISBN: 9781285195698
Textbook Problem
1 views

# In Exercises15 to 19, create drawing as needed. Given: Square A B C D with diagonals D B ¯ and A C ¯ intersecting at E D C = 5 3 Find: D B

To determine

To find:

DB of a square ABCD with diagonals DB¯ and AC¯ intersecting at E and DC=53.

Explanation

Approach:

For a right triangle for which the measure of the interior angles 45°, 45°, and 90°, if ‘a’ is the length of measure of one of the leg; opposite to the angle 45°, 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 =2× (Length of one of the legs)

Calculation:

Given,

ABCD is a square DB¯ and AC¯ intersecting at E and DC=53..

By the property of a square that all the sides and diagonals are of equal in length and the diagonals bisect each other at right angle, we have

AB¯BC¯CD¯DA¯,

AE¯BE¯CE¯DE¯,

DB¯AC¯, and

DBAC.

Thus,

Now, consider the right triangle ADE of the square ABCD, having right angle at E.

Since, two of the sides of the right triangle are congruent, the angles opposite to the sides must be equal and should be 45° each.

By the application of 45°-45°-90°

45°-45°-90° theorem.

In a right triangle whose angle measures 45°, 45°, and 90°, 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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