Question

at which store should michael buy his new smart phone?

1. triangle xyz is similar to triangle pqr

which proportions can be used to find the length of \\( \overline { p r } \\) in inches?
select two correct answers.

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, $PR$ is unknown.

Step2: Analyze Each Proportion

- **First Proportion: $\boldsymbol{\frac{8}{32}=\frac{PR}{17}}$**
Check corresponding sides: $YZ = 8$, $QR = 32$ (corresponding), $XZ = 17$, $PR$ (corresponding). So $\frac{YZ}{QR}=\frac{XZ}{PR} \implies \frac{8}{32}=\frac{17}{PR}$? Wait, no—wait, $\frac{YZ}{QR}=\frac{XZ}{PR}$ would be $\frac{8}{32}=\frac{17}{PR}$, but the option is $\frac{8}{32}=\frac{PR}{17}$. Wait, maybe I mixed up. Wait, $YZ = 8$, $QR = 32$ (ratio $\frac{8}{32}$), $XZ = 17$, $PR$ (ratio $\frac{17}{PR}$)? No, wait, similar triangles: $\frac{XY}{PQ}=\frac{YZ}{QR}=\frac{XZ}{PR}$. So $\frac{15}{60}=\frac{8}{32}=\frac{17}{PR}$. Let's check each option:
- Option 1: $\frac{8}{32}=\frac{PR}{17}$ → Cross-multiply: $8 \times 17 = 32 \times PR$ → $PR = \frac{136}{32} = 4.25$? No, that's wrong. Wait, no—wait, maybe $YZ$ (8) corresponds to $QR$ (32), and $XZ$ (17) corresponds to $PR$. So $\frac{YZ}{QR}=\frac{XZ}{PR} \implies \frac{8}{32}=\frac{17}{PR}$ → but the option is $\frac{8}{32}=\frac{PR}{17}$, which is $\frac{8}{32}=\frac{PR}{17} \implies PR = \frac{8 \times 17}{32} = 4.25$, which is incorrect. Wait, maybe I flipped. Wait, $\frac{QR}{YZ}=\frac{PR}{XZ}$ → $\frac{32}{8}=\frac{PR}{17}$ → $PR = 68$, which is correct (since $15/60 = 1/4$, so $PR$ should be $17 \times 4 = 68$). Wait, maybe the first option is $\frac{8}{32}=\frac{17}{PR}$? No, the option is $\frac{8}{32}=\frac{PR}{17}$. Wait, maybe I made a mistake. Let's check the fifth option: $\frac{60}{15}=\frac{PR}{17}$.

- **Fifth Proportion: $\boldsymbol{\frac{60}{15}=\frac{PR}{17}}$**
$PQ = 60$, $XY = 15$ (corresponding, ratio $\frac{60}{15}=4$), $XZ = 17$, $PR$ (ratio $\frac{PR}{17}$). So $\frac{PQ}{XY}=\frac{PR}{XZ} \implies \frac{60}{15}=\frac{PR}{17}$ → $4 = \frac{PR}{17}$ → $PR = 68$, which is correct (since $15 \times 4 = 60$, $17 \times 4 = 68$).

- **Third Proportion: $\boldsymbol{\frac{15}{60}=\frac{PR}{8}}$**
$XY = 15$, $PQ = 60$ (ratio $\frac{15}{60}$), $XZ = 17$ (no, $PR$ and $YZ = 8$? No, $YZ = 8$ corresponds to $QR = 32$, not $PR$. So this is wrong.

- **Fourth Proportion: $\boldsymbol{\frac{15}{17}=\frac{PR}{60}}$**
$XY = 15$, $XZ = 17$ (not corresponding), $PR$ and $PQ = 60$ (not corresponding). Wrong.

- **Fifth Proportion: $\boldsymbol{\frac{60}{15}=\frac{PR}{17}}$**
As above, $\frac{PQ}{XY}=\frac{PR}{XZ} \implies \frac{60}{15}=\frac{PR}{17}$ → $PR = 68$, correct.

Wait, maybe I messed up the first option. Wait, let's re-express the ratios:
$\frac{YZ}{QR}=\frac{XZ}{PR} \implies \frac{8}{32}=\frac{17}{PR}$ → but the option is $\frac{8}{32}=\frac{PR}{17}$, which is equivalent to $\frac{PR}{17}=\frac{8}{32}$ → cross-multiplying: $PR \times 32 = 8 \times 17$ → $PR = \frac{136}{32} = 4.25$, which is wrong. Wait, no—wait, maybe $XZ$ (17) corresponds to $PR$, and $YZ$ (8) corresponds to $QR$ (32). So $\frac{XZ}{PR}=\frac{YZ}{QR} \implies \frac{17}{PR}=\frac{8}{32}$ → which is $\frac{8}{32}=\frac{17}{PR}$, but the option is $\frac{8}{32}=\frac{PR}{17}$, which is the reciprocal. Wait, maybe the first option is correct if we consider $\frac{YZ}{QR}=\frac{PR}{XZ}$? No, that would be incorrect. Wait, maybe I made a mistake in correspondence. Let's list the sides:
$\triangle XYZ$: $XY = 15$ (vertical), $YZ = 8$ (horizontal), $XZ = 17$ (hypotenuse).
$\triangle PQR$: $PQ = 60$ (vertical), $QR = 32$ (horizontal), $PR$ (hypotenuse).
So vertical sides: $XY = 15$, $PQ = 60$ (ratio $\frac{15}{60} = \frac{1}{4}$).
Horizontal sides: $YZ = 8$, $QR = 32$ (ratio $\frac{8}{32} = \frac{1}{4}$).
Hypotenuses: $XZ = 17$, $PR$ (ratio $\frac{17}{PR} = \frac{1}{4}$ → $PR = 68$).

Now check each option:
1. $\frac{8}{32}=\frac{PR}{17}$ → $\frac{1}{4}=\frac{PR}{17}$ → $PR = \frac{17}{4} = 4.25$ (wrong).
2. $\frac{8}{32}=\frac{17}{PR}$ → $\frac{1}{4}=\frac{17}{PR}$ → $PR = 68$ (correct, but this option is written as $\frac{8}{32}=\frac{17}{PR}$? Wait, the second option is $\frac{8}{32}=\frac{17}{PR}$? Wait, the user's options: " $\frac{8}{32}=\frac{17}{PR}$ "? Wait, the original options:
- First: $\frac{8}{32}=\frac{PR}{17}$
- Second: $\frac{8}{32}=\frac{17}{PR}$
- Third: $\frac{15}{60}=\frac{PR}{8}$
- Fourth: $\frac{15}{17}=\frac{PR}{60}$
- Fifth: $\frac{60}{15}=\frac{PR}{17}$

Ah! I misread the second option. The second option is $\frac{8}{32}=\frac{17}{PR}$, which is correct (since $\frac{YZ}{QR}=\frac{XZ}{PR} \implies \frac{8}{32}=\frac{17}{PR}$).

- Fifth option: $\frac{60}{15}=\frac{PR}{17}$ → $\frac{PQ}{XY}=\frac{PR}{XZ} \implies \frac{60}{15}=\frac{PR}{17}$ → $4 = \frac{PR}{17}$ → $PR = 68$ (correct).

So the two correct proportions are:
- $\frac{8}{32}=\frac{17}{PR}$ (second option)
- $\frac{60}{15}=\frac{PR}{17}$ (fifth option)

Wait, but let's confirm:
For $\frac{8}{32}=\frac{17}{PR}$: Cross-multiply $8 \times PR = 32 \times 17$ → $PR = \frac{544}{8} = 68$ (correct).
For $\frac{60}{15}=\frac{PR}{17}$: Cross-multiply $60 \times 17 = 15 \times PR$ → $PR = \frac{1020}{15} = 68$ (correct).

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)
• $\boldsymbol{\frac{60}{15}=\frac{PR}{17}}$ (fifth option)

(Note: Based on the original options, the second option is $\frac{8}{32}=\frac{17}{PR}$ and the fifth is $\frac{60}{15}=\frac{PR}{17}$.)