What is the length of segment lm units

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Question:

Find the length of {eq}\overline{LM} {/eq}. Show your work.

Triangles:

Triangles are figures that have three sides that are connected through vertices. The vertices form internal angles, whose sum is equal to 180°. Some of the types of triangles are equilateral, isosceles, and scalene. We can use the law of sines to find an angle or a side of the triangle depending on the information we are given. The law of sines is the relationship between the sides and angles of a non-right triangle.

Answer and Explanation: 1

We are given a figure with a triangle. We are required to find the length of {eq}\overline{LM} {/eq}.

First, to find the length of {eq}\overline{LM} {/eq}, we can apply the law of sines:

$$\dfrac{\overline{LM} }{\sin N }=\dfrac{\overline{LN}}{\sin M} $$

Since we do not have the measure of the angle N, we can find it by applying that the sum of the internal angles is equal to {eq}180^{\circ} {/eq}. So:

$$\angle L + \angle N+ \angle M=180^{\circ} $$

Substituting the measures of the angles L and M, and solving, we have:

$$\begin{align} 53^{\circ} + \angle N+ 44^{\circ}&=180^{\circ} \\[0.3cm] \angle N+ 97^{\circ}-97^{\circ}&=180^{\circ} -97^{\circ} \\[0.3cm] \angle N &= 83^{\circ} \end{align} $$

Substituting {eq}\angle N=83^{\circ} {/eq}, {eq}\angle M=44^{\circ} {/eq}, {eq}\overline{LN}=14 {/eq} in the equation of the law of sines:

$$\dfrac{\overline{LM} }{\sin 83^{\circ} }=\dfrac{14}{\sin 44^{\circ}} $$

Solving for {eq}\overline{LM} {/eq}, we have:

$$\begin{align} \dfrac{\overline{LM} }{\sin 83^{\circ} }\times \sin 83^{\circ} &=\dfrac{14}{\sin 44^{\circ}}\times \sin 83^{\circ} \\[0.3cm] \overline{LM} &=20.153792\times \sin 83^{\circ} \\[0.3cm] \overline{LM} &\approx 20 \end{align} $$

Therefore, the length of the LM segment is approximately 20 units.

Learn more about this topic:

Properties of Shapes: Triangles

from

Chapter 14 / Lesson 2

Discover and define the angle properties of triangles. Learn about the different types of triangles based on sides and angles.

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