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:
[tex]x\left(x+7\right)=12^2[/tex]
solve the equation to find (x):
[tex]\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}[/tex]
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:
[tex]\begin{gathered} y^2=x^2+12^2 \\ y^2=9^2+12^2=81+144=225 \\ y=\sqrt{225}=15 \end{gathered}[/tex]
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:
[tex]\begin{gathered} z^2=12^2+16^2=144+256=400 \\ z=\sqrt{400}=20 \end{gathered}[/tex]
So, the answer will be:
[tex]\begin{gathered} x=9 \\ y=15 \\ z=20 \end{gathered}[/tex]
