Question

Explain why the triangles are similar and then find each length. please.

Answer 1

Since segments ST and VW are parallel, triangles VPW and SPT are similar. This is due to AAA (angle-angle-angle) theorem. In order to see this, we can draw the following picture

where we can see that angle V and angle S are the same, angle W and angle T are the same and angle P is the same in both triangles.

Now, since the triangles are similar, the following ratio must be preserved:

`(PS)/(10)=(PS+6)/(17.5)`

If we move 10 to the right hand side and 17.5 to the left hand side, we get

`17.5\cdot PS=10\cdot(PS+6)`

which is equal to

`17.5PS=10PS+60`

If we move 10PS to the left hand side as -10PS, we obtain

`\begin{gathered} 17.5PS-10PS=60 \\ 7.5PS=60 \\ PS=(60)/(7.5) \\ PS=8 \end{gathered}`

Now, since PT has the same length as PS and TW has the same length as SV, the answer is

`\begin{gathered} PS=PT=8 \\ SV=TW=6 \end{gathered}`