Chapter 5.6, Problem 40E

### 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

# Given: Δ R S T with right ∠ R S T ; m ∠ R = 30 ∘ and S T = 6 ; ∠ R S T is trisected by S M → and S N → Find: T N , N M , and M R

To determine

To find:

TN,NM,MR.

Explanation

Given:

In a Î”RST with right âˆ RST, mâˆ R=30âˆ˜,ST=6 and âˆ RST is trisected by SMâ†’,SNâ†’ as shown below

Theorem used:

Angle bisector theorem:

If a ray bisects one angle of a triangle, then it divides the opposite side into segments whose lengths are proportional to the lengths of the two sides that form the bisected angle.

Calculation:

Consider a Î”RST with right âˆ RST.

Since the âˆ RST is trisected, which is a right angle, we get mâˆ TSN=30âˆ˜,mâˆ NSM=30âˆ˜,mâˆ MSR=30âˆ˜.

Given that mâˆ R=30âˆ˜

â‡’mâˆ MSR=mâˆ R

Therefore triangle RMS is an isosceles triangle with MR=MS.

From the figure,

mâˆ TSM=mâˆ TSN+mâˆ NSM=30âˆ˜+30âˆ˜=60âˆ˜

mâˆ T=90âˆ˜âˆ’mâˆ TRS=90âˆ˜âˆ’30âˆ˜=60âˆ˜

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