Hello, Alice!

You've done all the preliminary work.

. . The answers are waiting for you . . .

Triangles $\displaystyle PQR$ and $\displaystyle XYZ$ are such that:

. . $\displaystyle \begin{array}{ccc}\angle P &=& \angle X \\

\angle Q &=& \angle Z \\ XY &=& 3\text{cm} \\ YZ &=& 4\text{cm}\\ PQ &=& 7\text{cm} \\ PR &=& 12\text{cm} \end{array}$

Find the lengths of $\displaystyle XZ$ and $\displaystyle QR$.

Did you make a sketch like this?

Code:

P
*
* *
* *
* * X
7 * * 12 *
* * * *
* * * * 3
* * * *
Q * * * * * * * * * R Z * * * * * Y
4

The two triangles are similar; their sides are proportional.

Since $\displaystyle PR = 12$ and $\displaystyle XY = 3$,

. . we see that $\displaystyle \Delta PQR$ is *four* times as large as $\displaystyle \Delta XYZ$

Then: .$\displaystyle QR \:=\:4\cdot YZ \quad\Rightarrow\quad QR \:=\:4(4) \quad\Rightarrow\quad QR\:=\:16$

And: .$\displaystyle PQ \:=\:4\cdot XZ \quad\Rightarrow\quad 7 \:=\:4\cdot XZ \quad\Rightarrow\quad XZ \:=\:\tfrac{7}{4}$