79. if a rectangle measures 54 meters by 72 meters, what is the length, in meters, of the diagonal of the rectangle?
a. 48
b. 63
c. 90
d. 126
e. 252
80. for all positive integers x, y, and z, which of the following expressions is equivalent to (\frac{x}{y} cdot \frac{z}{y})?
a. (\frac{y cdot z}{x cdot z})
b. (\frac{y cdot x}{y cdot z})
c. (\frac{y cdot z}{x cdot y})
d. (\frac{x cdot z}{y cdot y})
e. (\frac{y + z}{y + z})
81. what is the slope - intercept form of (8x - y - 6 = 0)?
a. (y = - 8x - 6)
b. (y = - 8x + 6)
c. (y = 8x + 6)
d. (y = 8x - 6)
e. (y = 6x - 8)
82. for right triangle (rst) shown below, what is ( an r)?
(there is a right triangle (rst) with right angle at (s), (rs = t), (st = r), (rt = s))
a. (\frac{r}{t})
b. (\frac{t}{r})
c. (\frac{r}{s})
d. (\frac{t}{s})
e. (\frac{s}{t})

Question

79. if a rectangle measures 54 meters by 72 meters, what is the length, in meters, of the diagonal of the rectangle?
a. 48
b. 63
c. 90
d. 126
e. 252
80. for all positive integers x, y, and z, which of the following expressions is equivalent to (\frac{x}{y} cdot \frac{z}{y})?
a. (\frac{y cdot z}{x cdot z})
b. (\frac{y cdot x}{y cdot z})
c. (\frac{y cdot z}{x cdot y})
d. (\frac{x cdot z}{y cdot y})
e. (\frac{y + z}{y + z})
81. what is the slope - intercept form of (8x - y - 6 = 0)?
a. (y = - 8x - 6)
b. (y = - 8x + 6)
c. (y = 8x + 6)
d. (y = 8x - 6)
e. (y = 6x - 8)
82. for right triangle (rst) shown below, what is (	an r)?
(there is a right triangle (rst) with right angle at (s), (rs = t), (st = r), (rt = s))
a. (\frac{r}{t})
b. (\frac{t}{r})
c. (\frac{r}{s})
d. (\frac{t}{s})
e. (\frac{s}{t})

Accepted Answer

### Problem 79:
# Explanation:
## Step1: Recall the Pythagorean theorem for a rectangle's diagonal.
For a rectangle with length $ l $ and width $ w $, the diagonal $ d $ is given by $ d = \sqrt{l^2 + w^2} $. Here, $ l = 72 $ meters and $ w = 54 $ meters.
## Step2: Calculate $ l^2 + w^2 $.
$ 72^2 = 5184 $ and $ 54^2 = 2916 $. Then $ 5184 + 2916 = 8100 $.
## Step3: Take the square root of the result.
$ \sqrt{8100} = 90 $.
# Answer:
90

### Problem 80:
# Explanation:
We start with $ \frac{x}{y}=\frac{z}{?} $. Cross - multiplying (if $ \frac{a}{b}=\frac{c}{d} $, then $ a\times d = b\times c $) gives us $ x\times?=y\times z $. So, $?=\frac{y\times z}{x} $. Let's check the options:
- Option A: $ \frac{y - z}{x} $ is not equal to $ \frac{yz}{x} $.
- Option B: $ \frac{y - x}{y} $ is not related.
- Option C: $ \frac{y - z}{y} $ is not related.
- Option D: $ \frac{z - y}{z} $ is not related.
- Option E: From $ \frac{x}{y}=\frac{z}{y + z} $? No, cross - multiplying $ \frac{x}{y}=\frac{z}{w} $ gives $ xw=yz$, if we solve for $ w $, $ w = \frac{yz}{x} $. Wait, maybe I misread. Wait, the question is "For all positive integers $ x $, $ y $, and $ z $, which of the following is equivalent to $ \frac{x}{y}=\frac{z}{?} $". Let's do cross - multiplication. If $ \frac{x}{y}=\frac{z}{k} $, then $ xk = yz$, so $ k=\frac{yz}{x} $. Wait, maybe the options are miswritten? Wait, maybe the original is $ \frac{x}{y}=\frac{z}{?} $, let's check the options again. Wait, option C is $ \frac{y - z}{y} $? No, maybe a typo. Wait, if we have $ \frac{x}{y}=\frac{z}{w} $, then $ w=\frac{yz}{x} $. None of the options seem to match? Wait, maybe the question is $ \frac{x}{y}=\frac{z}{y + z} $? No. Wait, maybe I made a mistake. Wait, let's re - express. Let's take the reciprocal? No. Wait, maybe the question is $ \frac{x}{y}=\frac{z}{y - z} $? No. Wait, maybe the options are different. Wait, the options are:
A. $ \frac{y - z}{x} $
B. $ \frac{y - x}{y} $
C. $ \frac{y - z}{y} $
D. $ \frac{z - y}{z} $
E. $ \frac{y - z}{y + z} $
Wait, maybe the question is $ \frac{x}{y}=\frac{z}{y - z} $? No, cross - multiplying $ \frac{x}{y}=\frac{z}{w} $ gives $ xw = yz$, so $ w=\frac{yz}{x} $. Since none of the options is $ \frac{yz}{x} $, maybe there is a misprint. But if we assume that the question is $ \frac{x}{y}=\frac{z}{y - z} $ is wrong. Wait, maybe the original question is $ \frac{x}{y}=\frac{z}{y + z} $ is wrong. Wait, maybe I misread the question. Wait, the question says "For all positive integers $ x $, $ y $, and $ z $, which of the following is equivalent to $ \frac{x}{y}=\frac{z}{?} $". Let's try option C: $ \frac{y - z}{y} $? No. Wait, maybe the question is $ \frac{x}{y}=\frac{z}{y} \times \frac{y}{x} $? No. I think there might be a mistake in the problem statement, but if we go with cross - multiplication, and assume that the answer is supposed to be $ \frac{yz}{x} $, but since it's not in the options, maybe I made a mistake. Wait, maybe the question is $ \frac{x}{y}=\frac{z}{y - z} $ is incorrect. Alternatively, maybe the question is $ \frac{x}{y}=\frac{z}{y + z} $ is incorrect. Wait, maybe the options are labeled wrong. If we consider that the correct answer is $ \frac{yz}{x} $, but since it's not there, maybe the intended answer is C? No. Wait, maybe the question is $ \frac{x}{y}=\frac{z}{y - z} $, cross - multiplying gives $ x(y - z)=yz$, $ xy-xz = yz$, $ xy=yz + xz$, $ xy=z(x + y) $, $ \frac{x}{y}=\frac{z}{x + y} $? No. I think there is a problem with the question, but if we have to choose, maybe the intended answer is C? No, I'm confused. Wait, maybe the original question is $ \frac{x}{y}=\frac{z}{y - z} $, then cross - multiplying $ x(y - z)=yz $, $ xy-xz = yz $, $ xy=z(x + y) $, $ \frac{x}{y}=\frac{z}{x + y} $, which is not an option. So maybe the question has a typo. But if we assume that the answer is C, but I'm not sure. Wait, maybe the question is $ \frac{x}{y}=\frac{z}{y} $, then $? = y $, but that's not an option. I think there is an error in the problem, but if we proceed, maybe the answer is C. (Note: This is a bit confusing due to possible typographical errors in the problem statement)
# Answer:
C (assuming due to possible typo in the problem)

### Problem 81:
# Explanation:
The slope - intercept form of a line is $ y=mx + b $, where $ m $ is the slope and $ b $ is the y - intercept. We start with the equation $ 8x-y - 6=0 $.
## Step1: Solve for $ y $.
Add $ y $ to both sides: $ 8x-6=y $, or $ y = 8x-6 $.
# Answer:
H (assuming option H is $ y = 8x-6 $)

### Problem 82:
# Explanation:
In right triangle $ RST $ with right angle at $ S $, $ \tan R=\frac{\text{opposite}}{\text{adjacent}} $ with respect to angle $ R $. The side opposite to angle $ R $ is $ ST $ (length $ r $) and the side adjacent to angle $ R $ is $ RS $ (length $ t $)? Wait, no. Wait, in right triangle $ RST $, right - angled at $ S $, angle $ R $: the opposite side is $ ST $ (length $ r $) and the adjacent side is $ RS $ (length $ t $)? Wait, the triangle has vertices $ R $, $ S $, $ T $, right - angled at $ S $. So $ RS $ and $ ST $ are the legs, and $ RT $ is the hypotenuse. So $ \tan R=\frac{\text{opposite}}{\text{adjacent}}=\frac{ST}{RS}=\frac{r}{t} $. But we need to know the labels. Wait, the triangle is labeled with $ RS = t $, $ ST = r $, $ RT = s $. So angle at $ R $: the opposite side is $ ST $ (length $ r $) and adjacent side is $ RS $ (length $ t $). So $ \tan R=\frac{r}{t} $. But we need to see the options. The options are $ F.\frac{r}{s} $, $ G.\frac{t}{r} $, $ H.\frac{r}{t} $, $ J.\frac{t}{s} $, $ K.\frac{s}{t} $. So the answer should be $ H $ (assuming $ \tan R=\frac{\text{opposite}}{\text{adjacent}}=\frac{ST}{RS}=\frac{r}{t} $).
# Answer:
H

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QUESTION IMAGE

Question

Explanation:

Problem 79:

Step1: Recall the Pythagorean theorem for a rectangle's diagonal.

Step2: Calculate \( l^2 + w^2 \).

Step3: Take the square root of the result.

Step1: Solve for \( y \).

Answer:

Problem 80: