Question

a triangle has side lengths of (q + r) centimeters, (5q − 10s) centimeters, and (5s − 7r) centimeters. which expression represents the perimeter, in centimeters, of the triangle? answer −16rs + 11qs −2rs + 2qr − 5qs 6q − 6r − 5s 6q − 2s − 9r submit answer

Explanation:

Step1: Recall perimeter formula

The perimeter \( P \) of a triangle is the sum of its three side lengths. So, \( P=(q + r)+(5q - 10s)+(5s - 7r) \).

Step2: Combine like terms for \( q \)

Combine the terms with \( q \): \( q+5q = 6q \).

Step3: Combine like terms for \( r \)

Combine the terms with \( r \): \( r-7r=-6r \)? Wait, no, wait. Wait, let's re - expand the sum:
\( (q + r)+(5q - 10s)+(5s - 7r)=q + r+5q-10s + 5s-7r \)
Now, combine \( r \) terms: \( r-7r=-6r \)? Wait, no, wait, let's do it again. Wait, \( r-7r=-6r \)? Wait, no, \( r-7r=(1 - 7)r=-6r \)? Wait, but let's check the \( s \) terms: \( - 10s+5s=-5s \)? Wait, no, wait, the last option is \( 6q-2s - 9r \)? Wait, I must have made a mistake. Let's re - calculate:

\( (q + r)+(5q - 10s)+(5s - 7r)=q+5q+r-7r-10s + 5s \)

For \( q \): \( q + 5q=6q \)

For \( r \): \( r-7r=(1 - 7)r=-6r \)? Wait, no, wait, \( r-7r=-6r \)? But the option has \( - 9r \). Wait, no, I think I messed up the signs. Wait, the third side is \( 5s-7r \), so when we expand:

\( (q + r)+(5q-10s)+(5s - 7r)=q + r+5q-10s+5s-7r \)

Now, \( r-7r=(1 - 7)r=-6r \)? Wait, no, \( 1r-7r=-6r \), but the option \( 6q-2s - 9r \) has \( - 9r \). Wait, maybe I made a mistake in the problem. Wait, no, let's check again:

Wait, \( (q + r)+(5q-10s)+(5s - 7r)=q+5q+r-7r-10s + 5s=6q+(r - 7r)+(-10s + 5s)=6q-6r-5s \)? No, that's one of the options, but there is another option \( 6q-2s - 9r \). Wait, maybe I misread the problem. Wait, the third side is \( 5s-7r \)? Wait, no, maybe it's \( 5s - 7r \)? Wait, let's re - add:

\( (q + r)+(5q-10s)+(5s - 7r)=q + 5q+r-7r-10s + 5s=6q-6r-5s \)? But that's option C (assuming the options are: first row: \( - 16rs + 11qs \), \( - 2rs+2qr - 5qs \); second row: \( 6q - 6r-5s \), \( 6q-2s - 9r \)). Wait, maybe I made a mistake. Wait, let's check the arithmetic again:

\( q+5q = 6q \)

\( r-7r=(1 - 7)r=-6r \)

\( - 10s+5s=-5s \)

So the perimeter is \( 6q-6r - 5s \)? But there is another option \( 6q-2s - 9r \). Wait, maybe the third side is \( 5s- 2r \)? No, the problem says \( 5s - 7r \). Wait, maybe I made a mistake in combining terms. Wait, no, \( r-7r=-6r \), \( - 10s + 5s=-5s \), so the sum is \( 6q-6r - 5s \). But let's check the other option: \( 6q-2s - 9r \). Let's see, if the third side was \( 5s-2r \), then \( r-2r=-r \), \( - 10s + 5s=-5s \), no. Wait, maybe the second side is \( 5q-2s \)? No, the problem says \( 5q - 10s \). Wait, maybe I misread the problem. Wait, the problem says: side lengths \( (q + r) \), \( (5q-10s) \), and \( (5s - 7r) \). So adding them:

\( (q + r)+(5q-10s)+(5s - 7r)=q + 5q+r-7r-10s + 5s=6q-6r-5s \). So the answer should be \( 6q - 6r-5s \). Wait, but let's check the options again. The options are:

1. \( - 16rs+11qs \)

1. \( - 2rs + 2qr-5qs \)

1. \( 6q-6r - 5s \)

1. \( 6q-2s - 9r \)

So the correct one is \( 6q-6r - 5s \).

Answer:

\( 6q - 6r-5s \)