Question

given: $overleftrightarrow{qs} perp overleftrightarrow{us}$ and $\triangle qrs sim \triangle stu$
prove: slope of $overleftrightarrow{qs}$ × slope of $overleftrightarrow{us} = -1$

statements	reasons
2. $\frac{qr}{st} = \frac{rs}{tu}$	property of similar triangles
3. slope of $overleftrightarrow{qs} = \frac{qr}{rs}$<br>slope of $overleftrightarrow{us} = -\frac{tu}{st}$	definition of slope
4. slope of $overleftrightarrow{qs}$ × slope of $overleftrightarrow{us} = \frac{qr}{rs} × -\frac{tu}{st}$	multiplying the slopes
5.	substitution property of equality
6. slope of $overleftrightarrow{qs}$ × slope of $overleftrightarrow{us} = -1$	simplifying the right side

the table shows the proof of the relationship between the slopes of two perpendicular lines. what is the missing statement in step 5?

a. slope of $overleftrightarrow{qs}$ × slope of $overleftrightarrow{us} = \frac{qr}{st} × \frac{rs}{tu}$

b. slope of $overleftrightarrow{qs}$ × $(-$slope of $overleftrightarrow{us}) = 1$

c. slope of $overleftrightarrow{qs}$ × slope of $overleftrightarrow{us} = 1$

d. slope of $overleftrightarrow{qs}$ × slope of $overleftrightarrow{us} = \frac{qr}{rs} × -\frac{rs}{qr}$

Explanation:

Step1: Recall Similar Triangles Property

From step 2, we know \(\frac{QR}{RS}=\frac{ST}{TU}\) (property of similar triangles). From step 3, slope of \(\overleftrightarrow{QS}=\frac{QR}{RS}\) and slope of \(\overleftrightarrow{US}=-\frac{TU}{ST}\) (definition of slope, considering direction).

Step2: Analyze Step 4 and Substitution

Step 4 is \(\text{slope of } \overleftrightarrow{QS} \times \text{slope of } \overleftrightarrow{US} = \frac{QR}{RS} \times -\frac{TU}{ST}\). From step 2, \(\frac{QR}{RS}=\frac{ST}{TU}\) (or \(\frac{QR}{ST}=\frac{RS}{TU}\), but let's use the first). Substitute \(\frac{QR}{RS}=\frac{ST}{TU}\) into the right - hand side: \(\frac{QR}{RS} \times -\frac{TU}{ST}=-\frac{QR \times TU}{RS \times ST}\). But since \(\frac{QR}{RS}=\frac{ST}{TU}\), we can rewrite \(\frac{QR}{RS} \times -\frac{TU}{ST}\) as \(\frac{ST}{TU} \times -\frac{TU}{ST}\) (by substituting \(\frac{QR}{RS}\) with \(\frac{ST}{TU}\) from step 2). Wait, alternatively, looking at the options, option D (assuming the last option is D, let's check the options again. The options are:

A. \(\text{slope of } \overleftrightarrow{QS} \times \text{slope of } \overleftrightarrow{US} = \frac{QR}{ST} \times \frac{RS}{TU}\)

B. \(\text{slope of } \overleftrightarrow{QS} \times (-\text{slope of } \overleftrightarrow{US}) = 1\)

C. \(\text{slope of } \overleftrightarrow{QS} \times \text{slope of } \overleftrightarrow{US} = 1\)

D. \(\text{slope of } \overleftrightarrow{QS} \times \text{slope of } \overleftrightarrow{US} = \frac{QR}{RS} \times -\frac{RS}{QR}\) (assuming the last option has a typo and is related to the substitution from step 2. Wait, from step 2, \(\frac{QR}{RS}=\frac{ST}{TU}\), so \(\frac{TU}{ST}=\frac{RS}{QR}\) (taking reciprocals and cross - multiplying). So \(\frac{QR}{RS} \times -\frac{TU}{ST}=\frac{QR}{RS} \times -\frac{RS}{QR}\) (since \(\frac{TU}{ST}=\frac{RS}{QR}\) from \(\frac{QR}{RS}=\frac{ST}{TU}\) implies \(QR\times TU = RS\times ST\) implies \(\frac{TU}{ST}=\frac{RS}{QR}\)).

So step 5 should be the substitution of \(\frac{TU}{ST}\) with \(\frac{RS}{QR}\) (or vice - versa) using the property of similar triangles (step 2). So the missing statement in step 5 is \(\text{slope of } \overleftrightarrow{QS} \times \text{slope of } \overleftrightarrow{US} = \frac{QR}{RS} \times -\frac{RS}{QR}\) (which is option D, assuming the last option is D as per the format).

Wait, let's re - express:

From step 2: \(\frac{QR}{RS}=\frac{ST}{TU}\Rightarrow\frac{TU}{ST}=\frac{RS}{QR}\)

Step 4: \(\text{slope of } \overleftrightarrow{QS} \times \text{slope of } \overleftrightarrow{US}=\frac{QR}{RS}\times(-\frac{TU}{ST})\)

Substitute \(\frac{TU}{ST}=\frac{RS}{QR}\) into step 4:

\(\text{slope of } \overleftrightarrow{QS} \times \text{slope of } \overleftrightarrow{US}=\frac{QR}{RS}\times(-\frac{RS}{QR})\)

Which is the statement in option D (assuming the last option is D). So the missing statement in step 5 is this substitution.

Answer:

D. \(\text{slope of } \overleftrightarrow{QS} \times \text{slope of } \overleftrightarrow{US} = \frac{QR}{RS} \times -\frac{RS}{QR}\) (or the option with this expression, depending on the exact option label. If the last option is D, then D is the answer)