   Chapter 10.4, Problem 18E 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

Use the analytic method to decide what type of quadrilateral is formed when the midpoints of the consecutive sides of a parallelogram are joined by the line segments.

To determine

What type of quadrilateral is formed for the given condition?

Explanation

Given theorem is,

The midpoints of the consecutive sides of a parallelogram are joined by the line segments

The above figure shows the parallelogram ABCD.

P, Q, R and S are the midpoints of the sides of the parallelogram AB, BC, CD and AD respectively.

Now, joining the midpoints PQ, QR, RS and SP as shown in the above figure.

The coordinate of the quadrilateral ABCD is A(0, 0), B(2a, 0),C(a+b, c) and D(b,c).

The midpoint of AB =P=0+2a2,0+02

The midpoint of AB =P=2a2,0

The midpoint of AB =P=a,0

The midpoint of BC =Q=2a+a+b2,0+c2

The midpoint of BC =Q=3a+b2,c2

The midpoint of CD =R=a+b+b2,c+c2

The midpoint of CD =R=a+2b2,2c2

The midpoint of CD =R=a+2b2,c

The midpoint of AD =S=0+b2,0+c2

The midpoint of AD =S=b2,c2

Using distance formula,

PQ =3a+b2-a

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