Question

In the diagram below, we have $ST = 2TR$ and $PQ = SR = 20$. Find the length $UV$.

Answer 1

Answer:

12

Step-by-step explanation:

1. $ST=2TR$ and $SR=20,$ then by segment addition postulate

$ST+TR=SR\\ \\2TR+TR=SR\\ \\3TR=20\\ \\TR=\dfrac{20}{3}\\ \\ST=2\cdot \dfrac{20}{3}=\dfrac{40}{3}$

2. Consider triangles PQU and TSU. These triangles are similar by AA similarity theorem (triangles have congruent vertical angles PUQ and TUS and congruent alternate interior angles PQU and TSU). Similar triangles have proportional corresponding sides, so

$\dfrac{QU}{US}=\dfrac{PQ}{ST}=\dfrac{20}{\frac{40}{3}}=\dfrac{60}{40}=1.5$

$QU=1.5US$

3. Consider triangles QUV and QSR. These triangles are similar by AA similarity theorem. Similar triangles have proportional corresponding sides, so

$\dfrac{QU}{QS}=\dfrac{UV}{SR}\\ \\\dfrac{QU}{QU+US}=\dfrac{1.5US}{1.5US+US}=\dfrac{1.5}{2.5}=0.6$

so

$\dfrac{UV}{SR}=0.6\Rightarrow UV=0.6SR=0.6\cdot 20=12$

Answer 2

Answer:

12

Step-by-step explanation:

Because ST/SR = 2/3 and PQ = SR, we have UQ/US = PQ/ST = SR/ST = 3/2.Since UQ/US = 3/2, we have UQ/QS = 3/5.

We have triangle UQV is similar to triangle SQR by AA Similarity, so UV/SR = UQ/QS = 3/5. Therefore, we have UV = (3/5)SR = 12