Question

How do I find the values of x, y, and z?

Answer 1

Given the figure of a right-angle triangle

We will find the values of x, y, and z.

First, we will find the value of (x):

From the small right-angle triangles:

`x\left(x+7\right)=12^2`

solve the equation to find (x):

`\begin{gathered} x^2+7x-144=0 \\ \lparen x-9)\left(x+16\right)=0 \\ x-9=0\rightarrow x=9 \\ x+16=0\rightarrow x=-16 \end{gathered}`

The negative result will be rejected.

So, x = 9

Second, we will find the value of (y)

From the right-angle triangle that is on the left side.

The hypotenuse = y

And the legs of the triangle are = x and 12 = 9 and 12

Using the Pythagorean theorem to find y as follows:

`\begin{gathered} y^2=x^2+12^2 \\ y^2=9^2+12^2=81+144=225 \\ y=√(225)=15 \end{gathered}`

So, y = 15

Finally, we will find the value of (z):

From the right-angle triangle that is on the right side.

The hypotenuse = z

And the legs are = 12 and x + 7 = 12 and 16

Using the Pythagorean theorem to find z as follows:

`\begin{gathered} z^2=12^2+16^2=144+256=400 \\ z=√(400)=20 \end{gathered}`

So, the answer will be:

`\begin{gathered} x=9 \\ y=15 \\ z=20 \end{gathered}`