   Chapter 10.CR, Problem 34CR Elementary Geometry For College St...

7th Edition
Alexander + 2 others
ISBN: 9781337614085

Solutions

Chapter
Section Elementary Geometry For College St...

7th Edition
Alexander + 2 others
ISBN: 9781337614085
Textbook Problem

Prove the statements in Review Exercises 32 to 36 using analytic geometry.If the diagonals of a trapezoid are equal in length, then the trapezoid is an isosceles trapezoid.

To determine

The analytic proof for the given theorem “If the diagonals of a trapezoid are equal in length, then the trapezoid is an isosceles trapezoid”.

Explanation

Given theorem is,

If the diagonals of a trapezoid are equal in length, then the trapezoid is an isosceles trapezoid.

The above graph shows the trapezoid ABCD.

The coordinates of the trapezoid are A0, 0, Ba,0, Cd, c and D(b, c).

The diagonals AC and B of the trapezoid is joined as shown in the figure.

To prove:

If AC=BD, then the trapezoid is isosceles trapezoid.

Characteristic of isosceles trapezoid:

i. The length of the diagonal are should be equal in length

ii. The lower and the upper bases are should be in parallel.

In the trapezoid ABCD in which base angles are equal and therefore left and right side lengths are equal.

Hence, d=b-a

The lower and upper bases AB and CD are in parallel.

And the Length of the diagonal AC= length of the diagonal BD.

Using distance formula for finding length of the diagonal AC.

Where, A0, 0 and Cb, c

AC=(d-0)2+(c-0)2

Simplifying the above equation,

AC=(d)2+(c)2

Where, B

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