Question

A triangle is labeled △LMN , and the angles can be represented by the following expressions: ∠L=4x+10 , ∠M=2x , and ∠N=44° . Use the angle measures to list the side lengths in increasing order.(1 point)
Responses

LN<LM<MN
upper L upper N is less than upper L upper M is less than upper M upper N

ML>NL>MN
upper M upper L is greater than upper N upper L is greater than upper M upper N

MN>NL>LM
upper M upper N is greater than upper N upper L is greater than upper L upper M

LM<MN<NL

Answer 1

To find the side lengths in increasing order based on the angles of triangle \( \triangle LMN \), we first need to determine the measures of the angles \( \angle L \) and \( \angle M \).

Given:
\[
\angle L = 4x + 10
\]
\[
\angle M = 2x
\]
\[
\angle N = 44^\circ
\]

Using the fact that the sum of the angles in a triangle is \( 180^\circ \), we can set up the equation:
\[
\angle L + \angle M + \angle N = 180^\circ
\]
Substituting the values we have:
\[
(4x + 10) + 2x + 44 = 180
\]
Simplifying:
\[
6x + 54 = 180
\]
Subtracting 54 from both sides:
\[
6x = 126
\]
Dividing by 6:
\[
x = 21
\]

Now substituting \( x \) back into the expressions for the angles:
\[
\angle L = 4(21) + 10 = 84 + 10 = 94^\circ
\]
\[
\angle M = 2(21) = 42^\circ
\]
\[
\angle N = 44^\circ
\]

Thus, the angle measures are:
- \( \angle L = 94^\circ \)
- \( \angle M = 42^\circ \)
- \( \angle N = 44^\circ \)

Now, we rank these angles:
- \( \angle M = 42^\circ \)
- \( \angle N = 44^\circ \)
- \( \angle L = 94^\circ \)

In a triangle, the side lengths are opposite to their respective angles. Therefore, the order from the smallest angle to the largest angle gives us the increasing order of the side lengths:

- \( \angle M \) (42°) → side \( LN \)
- \( \angle N \) (44°) → side \( LM \)
- \( \angle L \) (94°) → side \( MN \)

Thus, the side lengths in increasing order are:
\[
LN < LM < MN
\]

The correct response is:
\[
LN < LM < MN
\]