Question

Please!!Find the value of each variable. Write theequations and solve showing ALL the work.If an answer is not a whole number, leaveit in simplest radical form.

Answer 1

The greater triangle and the smaller ones (the two triangles inside the original one, that share the side y) are similar triangles, then we can formulate the following expressions:

• For the larger triangle and the triangle on the left

`(z)/(9)=(5)/(z)`

From this equation, we can solve for z, to get:

`\begin{gathered} (z)/(9)* z=(5)/(z)* z \\ (z* z)/(9)=5*(z)/(z) \\ (z^2)/(9)=5*1 \\ (z^2)/(9)=5 \\ z^2=5*9 \\ z^2=45 \\ z=\sqrt[]{45} \\ z=3\sqrt[]{5} \end{gathered}`

Then, z equals 3√5

• Similarly, with the larger triangle and the one on the right:

`\begin{gathered} (x)/(9)=(4)/(x) \\ \end{gathered}`

From this expression, we can solve for x, like this:

`\begin{gathered} (x)/(9)=(4)/(x) \\ (x^2)/(9)=4 \\ x^2=4*9 \\ x^2=36 \\ x=\sqrt[]{36} \\ x=6 \end{gathered}`

Then, x equals 6

• With the triangles on the right and on the left:

`(y)/(4)=(5)/(y)`

Solving for y, we get:

`\begin{gathered} (y)/(4)=(5)/(y) \\ (y^2)/(4)=5 \\ y^2=5*4 \\ y^2=20 \\ y=\sqrt[]{20} \\ y=2\sqrt[]{5} \end{gathered}`

Then, y equals 2√5