   Chapter 10.4, Problem 10E 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 rhombus form a rectangle.

To determine

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

Explanation

Given theorem is,

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

The above figure shows the rhombus ABCD.

Length of all the side of the rhombus is equal and the opposite angles are equal.

AB=BC=CD=DA

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

Now, joining the midpoints ST, TU, UV and VS as shown in the above figure.

And AC and BD are the diagonals of the rhombus.

Considering the triangles VUC and SAT,

A=C

CV=AT and UC=SA

Based on the SAS property, the triangle VUC is congruent to the triangle SAT.

When two triangles are congruent, the corresponding sides and corresponding angle are same.

Hence, VU=ST.

Similarly, in triangle DSV and UTB

D=B

DS=BU and DV=BT

Based on the SAS property, the triangle DSV is congruent to the triangle UTB

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