Chapter 4.2, Problem 31E

### Elementary Geometry For College St...

7th Edition
Alexander + 2 others
ISBN: 9781337614085

Chapter
Section

### Elementary Geometry For College St...

7th Edition
Alexander + 2 others
ISBN: 9781337614085
Textbook Problem

# In Exercises 29 to 31, M and N are the midpoints of sides R S ¯ and R T ¯ of ∆ R S T , respectively. Given: R M = R N = 2 x + 1   S T = 5 x - 3   m ∠ R = 60 ° Find: x , R M , and S T

To determine

To find:

x, RM, and ST

For RM=RN=2x+1

ST=5x-3

mR=60°

Explanation

Calculation:

Given,

RM=RN=2x+1

ST=5x-3

mâˆ R=60Â°

M and N are the midpoints of sides RSÂ¯ and RTÂ¯ OF âˆ†RST respectively.

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

Thus,

MN=12ST

MN=125x-3

Now, in âˆ†RMN we have

mâˆ R=60Â°

But, RM=RN

Since, angle opposite to the equal sides are equal in measure,

mâˆ M=mâˆ N=xsay

By the angle sum property of a triangle in âˆ†RMN

âˆ R+âˆ M+âˆ N=18OÂ°

60Â°+x+x=180Â°

60Â°+2x=180Â°

60Â°+2x+-60Â°=180Â°+-60Â°

2x+60Â°+-60Â°=180Â°-60Â°

2x=120Â°

Divide; by 2 on both sides,

2x2=120Â°2

x=60Â°

Thus,

âˆ R=âˆ M=âˆ N=60Â°

Hence, âˆ†RMN is an equilateral triangle

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