Chapter 10.4, Problem 10E

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6th Edition

Daniel C. Alexander + 1 other

ISBN: 9781285195698

Textbook Problem

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*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

**To answer:**

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.

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,

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,

Similarly, in triangle DSV and UTB

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

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