   Chapter 4.3, Problem 42E 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

a) Argue that the midpoint of the hypotenuse of a right triangle is equidistant from the three vertices of the triangle. Use the fact that the congruent diagonals of a rectangle bisect each other. Be sure to provide a drawing.b) Use the relationship from part a) to find CM, the length of the median to the hypotenuse of right △ A B C , in which m ∠ C = 90 ∘ , AC = 6, and BC = 8.

To determine

(a)

To argue:

The midpoint of the hypotenuse of a right triangle is equidistant from the three vertices of the triangle.

Explanation

Calculation:

We have to use the fact that the congruent diagonals of a rectangle bisect each other.

Consider the rectangle ABCD whose congruent diagonals are AC and BD.

ABC is right triangle. Since it is a midpoint, clearly it is equidistant from the vertices that are opposite the 90 degree angle. Thus, we must prove that triangle AOB is isosceles. Since angle A and angle B are both bisected 90 degree angles, they have equal measures. Since the base angles are the same, it is isosceles.

ABC is right angled with hypotenuse = AC, as per the question

To determine

(b)

To find:

CM, the length of the median to the hypotenuse of right ABC.

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