Chapter 5.5, Problem 22E

### 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: Rhombus A B C D (not shown) in whichdiagonals A C ¯ and D B ¯ intersect at point E ; D B = A B = 8 Find: A C

To determine

To find:

AC of a rhombus ABCD in which the diagonals AC¯ and BD¯ intersect at the point E with BD=AB=8

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,

A rhombus ABCD with ACÂ¯ and BDÂ¯ as diagonals wjhich intersect at the point E, with BD=AB=8

By the property of a rhombus that all the four sides of a rhombus are congruent with that the diagonals are their perpendicular bisector we have,

ABÂ¯â‰…BCÂ¯â‰…CDÂ¯â‰…DAÂ¯,

AEÂ¯â‰…ECÂ¯; BEÂ¯â‰…EDÂ¯ and

ACÂ¯âŠ¥BDÂ¯

Thus,

AB=BC=CD=DA=8 units.

BE=DE=4 units

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