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 LM<MN<NL upper L upper M is less than upper M upper N is less than upper N upper L 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 LN<LM<MN
Answers
GPT-4o mini
To find the side lengths of triangle \( \triangle LMN \) in increasing order based on the angle measures, we first need to find the values of angles \( \angle L \), \( \angle M \), and \( \angle N \).
We know that the sum of the angles in a triangle is \( 180^\circ \). Therefore, we can set up the equation:
\[
\angle L + \angle M + \angle N = 180^\circ
\]
Substituting the expressions for the angles:
\[
(4x + 10) + (2x) + 44 = 180
\]
Combine like terms:
\[
6x + 54 = 180
\]
Now, solve for \( x \):
\[
6x = 180 - 54
\]
\[
6x = 126
\]
\[
x = 21
\]
Now that we have \( x \), we can substitute it back to find the measures of each angle:
1. Angle \( \angle L \):
\[
\angle L = 4x + 10 = 4(21) + 10 = 84 + 10 = 94^\circ
\]
2. Angle \( \angle M \):
\[
\angle M = 2x = 2(21) = 42^\circ
\]
3. Angle \( \angle N \):
\[
\angle N = 44^\circ
\]
Now we have:
\[
\angle L = 94^\circ, \quad \angle M = 42^\circ, \quad \angle N = 44^\circ
\]
According to the angle-side theorem, the sides opposite the larger angles are longer. The order of the angles from smallest to largest is:
\[
\angle M < \angle N < \angle L \quad (42^\circ < 44^\circ < 94^\circ)
\]
This means the sides opposite these angles will have the same order:
1. Side opposite \( \angle M \) (LM) is the shortest.
2. Side opposite \( \angle N \) (MN) is in the middle.
3. Side opposite \( \angle L \) (NL) is the longest.
Thus, the order of the side lengths in increasing order is:
\[
LM < MN < NL
\]
Therefore, the final answer is:
\[
\text{LM < MN < NL}
\]
We know that the sum of the angles in a triangle is \( 180^\circ \). Therefore, we can set up the equation:
\[
\angle L + \angle M + \angle N = 180^\circ
\]
Substituting the expressions for the angles:
\[
(4x + 10) + (2x) + 44 = 180
\]
Combine like terms:
\[
6x + 54 = 180
\]
Now, solve for \( x \):
\[
6x = 180 - 54
\]
\[
6x = 126
\]
\[
x = 21
\]
Now that we have \( x \), we can substitute it back to find the measures of each angle:
1. Angle \( \angle L \):
\[
\angle L = 4x + 10 = 4(21) + 10 = 84 + 10 = 94^\circ
\]
2. Angle \( \angle M \):
\[
\angle M = 2x = 2(21) = 42^\circ
\]
3. Angle \( \angle N \):
\[
\angle N = 44^\circ
\]
Now we have:
\[
\angle L = 94^\circ, \quad \angle M = 42^\circ, \quad \angle N = 44^\circ
\]
According to the angle-side theorem, the sides opposite the larger angles are longer. The order of the angles from smallest to largest is:
\[
\angle M < \angle N < \angle L \quad (42^\circ < 44^\circ < 94^\circ)
\]
This means the sides opposite these angles will have the same order:
1. Side opposite \( \angle M \) (LM) is the shortest.
2. Side opposite \( \angle N \) (MN) is in the middle.
3. Side opposite \( \angle L \) (NL) is the longest.
Thus, the order of the side lengths in increasing order is:
\[
LM < MN < NL
\]
Therefore, the final answer is:
\[
\text{LM < MN < NL}
\]