   Chapter 10.CR, Problem 32CR 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 segment that join the midpoints of consecutive sides of a parallelogram from another parallelogram.

To determine

The analytic proof for the given theorem “The line segment that join the midpoints of consecutive sides of a parallelogram from another parallelogram”

Explanation

Given theorem is,

The line segment that join the midpoints of consecutive sides of a parallelogram from another parallelogram

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-a2

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