Question

1. triangle xyz is similar to triangle pqr.

two right triangles: xyz with legs 15 in (xy), 8 in (yz), hypotenuse 17 in (xz); pqr with leg 32 in (qr), other leg 60 in (pq), hypotenuse pr (unknown).

which proportions can be used to find the length of pr in inches? select two correct answers.

options (some marked with x or check):

• \\(\frac{8}{32} = \frac{pr}{17}\\)
• \\(\frac{8}{32} = \frac{17}{pr}\\) (marked x?)
• \\(\frac{15}{60} = \frac{pr}{8}\\) (marked x)
• \\(\frac{15}{17} = \frac{pr}{60}\\) (marked x)
• \\(\frac{60}{15} = \frac{pr}{17}\\) (marked with check)

(other handwritten marks and a red 1/2 at bottom.)

Explanation:

Step1: Recall Similar Triangles Property

In similar triangles, corresponding sides are proportional. For $\triangle XYZ \sim \triangle PQR$, the corresponding sides are: $XY$ corresponds to $PQ$, $YZ$ corresponds to $QR$, and $XZ$ corresponds to $PR$.

So, $XY = 15$ in, $PQ = 60$ in; $YZ = 8$ in, $QR = 32$ in; $XZ = 17$ in, $PR =?$

Step2: Analyze Proportions

Proportion 1: $\frac{8}{32}=\frac{17}{PR}$

• Corresponding sides: $YZ$ (8) and $QR$ (32), $XZ$ (17) and $PR$. So, $\frac{YZ}{QR}=\frac{XZ}{PR} \implies \frac{8}{32}=\frac{17}{PR}$.

Proportion 2: $\frac{60}{15}=\frac{PR}{17}$

• Corresponding sides: $PQ$ (60) and $XY$ (15), $PR$ and $XZ$ (17). So, $\frac{PQ}{XY}=\frac{PR}{XZ} \implies \frac{60}{15}=\frac{PR}{17}$.

Now, check the given options:

• Option with $\frac{8}{32}=\frac{17}{PR}$ (matches the first proportion).
• Option with $\frac{60}{15}=\frac{PR}{17}$ (matches the second proportion, can also be written as $\frac{60}{15}=\frac{PR}{17}$ or rearranged, but the option $\frac{60}{15}=\frac{PR}{17}$ (or equivalent $\frac{60}{15}=\frac{PR}{17}$) is correct. Wait, the option given as $\frac{60}{15}=\frac{PR}{17}$ (or the one with $\frac{60}{15}=\frac{PR}{17}$) and $\frac{8}{32}=\frac{17}{PR}$.

From the options, the two correct proportions are:

1. $\frac{8}{32}=\frac{17}{PR}$ (second option in the list, assuming the options are as: first $\frac{8}{32}=\frac{PR}{17}$ (wrong), second $\frac{8}{32}=\frac{17}{PR}$ (correct), and $\frac{60}{15}=\frac{PR}{17}$ (fifth option, $\frac{60}{15}=\frac{PR}{17}$) and the fifth option $\frac{60}{15}=\frac{PR}{17}$ (or $\frac{60}{15}=\frac{PR}{17}$)). Wait, rechecking the options:

Wait the options are:

1. $\frac{8}{32}=\frac{PR}{17}$ (wrong, should be $\frac{17}{PR}$)
2. $\frac{8}{32}=\frac{17}{PR}$ (correct)
3. $\frac{15}{60}=\frac{PR}{8}$ (wrong)
4. $\frac{15}{17}=\frac{PR}{60}$ (wrong)
5. $\frac{60}{15}=\frac{PR}{17}$ (correct, since $\frac{PQ}{XY}=\frac{PR}{XZ} \implies \frac{60}{15}=\frac{PR}{17}$)

So the two correct options are the second one ($\frac{8}{32}=\frac{17}{PR}$) and the fifth one ($\frac{60}{15}=\frac{PR}{17}$).

Answer:

The two correct proportions are:

• $\boldsymbol{\frac{8}{32}=\frac{17}{PR}}$ (second option, assuming the options are ordered as given)
• $\boldsymbol{\frac{60}{15}=\frac{PR}{17}}$ (fifth option, assuming the options are ordered as given)

(Note: Based on the visual, the two correct options are the one with $\frac{8}{32}=\frac{17}{PR}$ and the one with $\frac{60}{15}=\frac{PR}{17}$ (or its equivalent $\frac{60}{15}=\frac{PR}{17}$).)