   Chapter 10.CR, Problem 36CR 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.The line segments joining the midpoints of consecutive sides of an isosceles trapezoid form a rhombus.

To determine

The analytic proof for the given theorem “The line segments joining the midpoints of consecutive sides of an isosceles trapezoid form a rhombus.”

Explanation

Given theorem is,

The line segments joining the midpoints of consecutive sides of an isosceles trapezoid from a rhombus.

The above graph shows the isosceles trapezoid ABCD.

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

Hence, d=b-a

P, Q, R and S are the midpoint of the sides of the isosceles trapezoid AB,BC, CD and AD respectively.

Midpoint of AB=P=a+02,0+02

Midpoint of AB=P=a2,0

Midpoint of BC=Q=a+d2,0+c2

Midpoint of BC=Q=a+d2,c2

Midpoint of CD=R=b+d2,c+c2

Midpoint of CD=R=b+d2,c

Midpoint of DA=S=b+02,0+c2

Midpoint of DA=S=b2,c2

Length of PQ=a+d2-a22+c2-02

Length of PQ=d22+c22

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