Question

TPretest: Unit 2In the figure, PO is parallel to RS°. The length of Po is 2 cm; the length ofRS is 6 cm; the length of o7 is 4 cm. What is the length of TS?

Answer 1

Explanation

First of all it's important to notice that the intersections at points P and R form the same groups of four angles because they are formed by a transversal segment (TR) intersecting two parallel segments (PQ and RS). The same happens at the intersections at Q and S.

Those observations imply the following equalities between angles:

$\begin{gathered} m\angle TPQ=m\angle TRS \\ m\angle PQT=m\angle RST \end{gathered}$

This means that triangles RST and PQT have the same angles and therefore according to the AAA similiraty rule they are similar. If they are similar then the following relations between corresponding sides of different triangles are met:

$\begin{gathered} TR=k\cdot TP \\ RS=k\cdot PQ \\ TS=k\cdot QT \end{gathered}$

Where k is a number known as the scale factor.

We have the lengths of RS, PQ, and QT so the second and third inequalities above can be written as:

$\begin{gathered} RS=k\cdot PQ\Rightarrow6=2k \\ TS=k\cdot QT\Rightarrow TS=4k \end{gathered}$

We can divide both sides of the first equation:

$\begin{gathered} \frac{6}{2}=\frac{2k}{2} \\ k=3 \end{gathered}$

Now that we have the value of the scale factor we can find the length of TS:

$TS=4\cdot3=12$Answer

Then the answer is option C, 12 cm.