   Chapter 10.4, Problem 33E 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

Would the theorem of Exercise 7 remain true for a concave quadrilateral like the one shown? Exercise 33.

To determine

The analytic proof for the given theorem “Would the theorem of Exercise 7 remain true for the given concave quadrilateral”

Explanation

Given figure is,

The above figure shows the quadrilateral.

And the theorem of Exercise 7 is “The line segments that join the midpoints of the consecutive sides of a quadrilateral form a parallelogram”

To prove “The line segments that join the midpoints of the consecutive sides of a quadrilateral form a parallelogram” for the given figure, follow as below,

The coordinates of the quadrilateral ABCD is as shown in the figure,

Let R, S, T, V are the midpoint of AB, BC, CD and AD.

Joining the R, T, TV and VR as shown in the figure.

Midpoint of AB = S = 7+02, 0+02

Midpoint of AB = S = 72,02

Midpoint of AB = S = 3.5, 0

Midpoint of BC = T = 7+62, 0+22

Midpoint of BC = T = 132,22

Midpoint of BC = T = 6.5, 1

Midpoint of CD = V = 8+62, 7+22

Midpoint of CD = V = 142,92

Midpoint of CD = V = 7, 4.5

Midpoint of AD = R = 8+02, 7+02

Midpoint of AD = R = 82,72

Midpoint of AD = R = 4, 3.5

Determining the slopes as below,

mST-=1-06

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