   Chapter 10.4, Problem 9E 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 rectangle form a rhombus.

To determine

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

Explanation

Given theorem is,

The line segments that join the midpoints of the consecutive sides of a rectangle form a rhombus.

The above figure shows the rectangle ABCD.

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

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

And AC and BD are the diagonals of the 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 VT

And VT=12AC

Similarly, In Triangle ABC,

AC is parallel to RS

And RS=12AC

Hence, VT=RS

Similarly when considering the triangles DAB and CDB

VR=TS

R, V and S are the midpoint of the sides AB, AD and BC

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