Question

Find the values of x, y, z, A, and B in the image below. List the facts that both triangles have 90° angles and that the triangles are similar. What is the value of A and B in degrees? What is the measure of y and z?

Answer 1

Answer:

$\begin{gathered} A=36.87\text{\degree} \\ B=53.13\text{\degree} \\ x=12 \\ y=10.15 \\ z=6.25 \end{gathered}$

Step-by-step explanation:

First, we'll work on the triangle that's on the left side. We'll find the values of A,B and x.

Using the law of sines, we'll have that:

$\frac{\sin(90)}{15}=\frac{\sin(A)}{9}$

Solving for A,

$\begin{gathered} \frac{\sin(90)}{15}=\frac{\sin(A)}{9} \\ \\ \rightarrow\sin(A)=\frac{9\sin(90)}{15} \\ \\ \rightarrow\sin(A)=\frac{3}{5} \\ \\ \rightarrow A=\sin^{-1}(\frac{3}{5}) \\ \\ \Rightarrow A=36.87\text{\degree} \end{gathered}$

Now, since we know that the sum of the interior angles of a triangle is 180°, we'll have that:

$\begin{gathered} 90+A+B=180 \\ \rightarrow B=180-A-90 \\ \rightarrow B=180-36.87-90 \\ \\ \Rightarrow B=53.13\text{\degree} \end{gathered}$

Using the law of sines again, we'll get that:

$\frac{x}{\sin(B)}=\frac{15}{\sin(90)}$

Solving for x,

$\begin{gathered} \frac{x}{\sin(B)}=\frac{15}{\sin(90)} \\ \\ \rightarrow x=\frac{15\sin(B)}{\sin(90)}\rightarrow x=\frac{15\sin(53.13)}{\sin(90)} \\ \\ \Rightarrow x=12 \end{gathered}$

Now, we'll work on the triangle that's on the right side. We'll find the values of y and z.

Since this is a right triangle (it has a 90° angle), we can say that:

$\cos(38)=\frac{8}{y}$

Solving for y,

$\begin{gathered} \cos(38)=\frac{8}{y}\rightarrow y\cos(38)=8\rightarrow y=\frac{8}{\cos(38)} \\ \\ \Rightarrow y=10.15 \end{gathered}$

We can also state that:

$\tan(38)=\frac{z}{8}$

Soving for z,

$\begin{gathered} \tan(38)=\frac{z}{8}\rightarrow z=8\tan(38) \\ \\ \Rightarrow z=6.25 \\ \end{gathered}$

This way, we can conclude that:

$\begin{gathered} A=36.87\text{\degree} \\ B=53.13\text{\degree} \\ x=12 \\ y=10.15 \\ z=6.25 \end{gathered}$