Question

In the diagram shown below, triangle JKN ~ triangle NKM ~ triangle MKL.What is the length, in units, of NM? Show your work. Explain your reasoning.thank you ! :)

Answer 1

Answer:

$NM=2\sqrt{3}\text{ }units$

Explanation:

We are given that the 3 triangles are similar. This means that the corresponding angles of each triangle are equal.

We are given the measure of the angle NMK = 60º. Since the triangle NKM is similar to triangle MKL, the angle MLK = 60º = NMK

We have the triangle MKL:

Now, we can use the trigonometric ratio sine, to find the length of MK:

$\sin(60º)=\frac{MK}{8}$

Thus:

$MK=8\sin(60º)=8\cdot\frac{\sqrt{3}}{2}=4\sqrt{3}$

And now, if we look at the triangle NKM:

And now, we can find the asked length, the length of side NM.

Using the trigonometric ratio cosine:

$\cos(60º)=\frac{NM}{4\sqrt{3}}$

And solve:

$NM=4\sqrt{3}\cos(60º)=4\sqrt{3}\cdot\frac{1}{2}=2\sqrt{3}$

Thus, the length of side NM is 4√3