   Chapter 10.4, Problem 8E 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 opposite sides of a quadrilateral bisect each other.

To determine

The analytic proof for the given theorem “The line segments that join the midpoints of the opposite sides of a quadrilateral bisect each other”.

Explanation

Given theorem is,

The line segments that join the midpoints of the opposite sides of a quadrilateral bisect each other.

The above figure shows the quadrilateral ABCD.

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

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

Moreover, AC and BD are the diagonals of the quadrilateral.

PR and QS are the line segments that join the midpoints of the opposite sides of a quadrilateral.

In the quadrilateral ABCD, DAC and ABC are the two triangle and based on the midpoint theorem,

In Triangle DAC,

AC is parallel to PS

And PS=12AC

Similarly, In Triangle ABC,

AC is parallel to QR

And QR=12AC

Hence, P

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