   Chapter 10.4, Problem 13E 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 segment that joins the midpoints of two sides of a triangle is parallel to the third side and has a length equal to one-half the length of the third side.

To determine

The analytic proof for the given theorem “The line segment that joins the midpoints of two sides of a triangle is parallel to the third side and has a length equal to one-half the length of the third side”.

Explanation

Given theorem is,

The line segment that joins the midpoints of two sides of a triangle is parallel to the third side and has a length equal to one-half the length of the third side.

The above figure shows the triangle ABC.

The vertices of the triangle ABC is A (0, 0), B (4a,0) and C (2a,4b).

D and E are the midpoints of the sides AC and BC.

Now, joining the points D and E as shown in the figure.

Hence, the line DE is parallel to the third side of the triangle AB.

Based on the midpoint formula,

The midpoint of AC =D=0+2a2,0+4b2

The midpoint of AC =D=a,2b

Similarly, midpoint of BC =E=4a+2a2,0+4b2

The midpoint of BC =E=3a,2b

Based on the distance formula, the length of DE which has the coordinates of D a,2b and E 3a,2b as below,

DE =(3a-a)2

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