In this exercise we are asked to find lengths of three segments. Let's take a look at the given picture.
First, let's notice that, since PR is parallel to WX, △ YPS and △ WQS are similar by Angle-Angle Similarity Theorem. Using the same theorem, we can prove that △ WQS and △ WXY are similar.
By the Transitive Property of Similarity, △ YPS is similar to △ WXY. This means that corresponding sides of these triangles are proportional.
PY/WX=PS/XY=SY/WY
Let's solve the above proportion by substituting appropriate side lengths. We will start with the left and the middle ratio.
The length of SY is 4. Let's add this information to our picture.
Having in mind the Transitive Property of Similarity, we can see that △ PQR is similar to △ XYW. Let's use the fact that in similar triangles corresponding sides are proportional.
PQ/XY=PR/XW
⇓
PQ/6=5+ 5/10
Finally, we will solve the above proportion to find the length of PQ.
