   Chapter 10.4, Problem 7E 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

In Exercises 1 to 17, complete an analytic proof for each theorem.The line segments that join the midpoints of the consecutive sides of a quadrilateral from a parallelogram.

To determine

The analytic proof for the given theorem “The line segments that join the midpoints of the consecutive sides of a quadrilateral form a parallelogram”.

Explanation

Given theorem is,

The line segments that join the midpoints of the consecutive sides of a quadrilateral form a parallelogram

The above figure shows the quadrilateral ABCD.

R, S, T and V are the midpoints of the ides of the quadrilateral AB, BC, CD and AD respectively.

Now, joining the midpoints RS, ST, TV and VR as shown in the above figure.

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

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

R=2a2,0=a,0

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

S=2(a+b)2,c=a+b,c

The midpoint of CD =T=2d+2b2,2e+2c2

T=2(d+b)2,2(e+c)2=d+b,e+c

The midpoint of AD =V=0+2d2,0+2e2

V=2d2,2e2=d,e

Determining the slopes as below,

mRS-=

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