Side-Side-Side Similarity Theorem: If the corresponding side lengths of two triangles are proportional, then the triangles are similar.
Side-Angle-Side Similarity Theorem: If the lengths of two sides of one triangle are proportional to the lengths of two corresponding sides of another triangle and the included angles are congruent, then the triangles are similar.
Now we will identify the similar triangles and find the measures, one at a time.
Similar Triangles
We want to identify the similar triangles in the given diagram.
Notice that ∠ SVQ is congruent to ∠ STR. We can also see that △ QSV and △ RST share ∠ S. This means that two angles of △ QSV are congruent to two angles of △ RST. Therefore, by the Angle-Angle Similarity Theorem, △ QSV and △ RST are similar.
△ QSV ~ △ RST
Finding the Measures
Using our similarity statement from above, we can identify two pairs of corresponding sides that will help us find the requested lengths.
VQ corresponds with TR
VS corresponds with TS
Recall that corresponding segments of similar figures will have proportional lengths. We are given expressions for the lengths of these sides which we can use to write a proportion.
VQ/TR = VS/TS
⇕
5/3 = x/12
Let's solve this equation to find x.
