   Chapter 10.CR, Problem 35CR 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

Prove the statements in Review Exercises 32 to 36 using analytic geometry.If two medians of a triangle are equal in length, then the triangle is isosceles.

To determine

The analytic proof for the given theorem “If two medians of a triangle are equal in length, then the triangle is isosceles”.

Explanation

Given theorem is,

If two medians of a triangle are equal in length, then the triangle is isosceles.

Condition of isosceles triangle AC=BCAB

The above graph shows the triangle ABC.

The coordinates of the triangle a0, 0, b4a, o, C2a, 2b).

In a triangle ABC, the side length of AC and BC are equal and congruent. That can be proved using distance formula as below,

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

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

Length of AC =4a2+4b2

Length of AC =4(a2+b2)

Length of AC =2(a2+b2)

Similarly the length BC as below,

Length of AC =(4a-2a)2+(2b-0)2

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

Length of AC =4a2+4b2

Length of AC =4(a2+b2)

Length of AC =2(a2+b2)

Similarly the length AB as below,

Length of AB =(4a-0)2+(0-0)2

Length of AB =(4a)2

Length of AB =16a2

Length of AB =4a

Hence, AC=BCAB is proved

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