   Chapter 10.4, Problem 16E ### Elementary Geometry for College St...

6th Edition
Daniel C. Alexander + 1 other
ISBN: 9781285195698

#### Solutions

Chapter
Section ### Elementary Geometry for College St...

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.If the median to one side of a triangle is also an altitude of the triangle, then the triangle is isosceles.

To determine

The analytic proof for the given theorem “If the median to one side of a triangle is also an altitude of the triangle, then the triangle is isosceles.”

Explanation

Given theorem is,

If the median to one side of a triangle is also an altitude of the triangle, then the triangle is isosceles.

The above graph shows the triangle ABC.

Condition for isosceles triangle:

In a triangle ABC, the side length of AC and BC are equal.

In the above figure AD is the median drawn from the vertex C to the midpoint of the base AB.

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

D is the midpoint of base AB and Joining CD as shown in the figure.

Here, CD is the median of the triangle ABC.

And, CD is an altitude of the triangle ABC because CD is perpendicular to the base line AB.

Hence, the median and altitude of the triangle ABC is CD.

D=4a+02,0+02

D=4a2,0

D=2a,0

Now, determining the length of AC, AB and BC to prove the triangle is isosceles as below,

Length of AC using distance formula,

AC=(2a-0)2+(2b-0)2

AC=(2a)2+(2b)2

AC=4a2+4b2

AC=4(a2+b2)

AC=2(a2+b2)

Length of BC using distance formula,

BC=(2a-4a)

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