Question

what is the area of parallelogram rstu?
21 square units
24 square units
28 square units
32 square units

Explanation:

Step1: Find the base length

From the coordinates, the distance between U(-3, -1) and S(4, -1) (or along the horizontal line) is the base. Using the distance formula for horizontal line (since y-coordinates are same), base \( b = |4 - (-3)| = 7 \)? Wait, no, wait. Wait, looking at the graph, actually, the vertical side: S is (4, -1) and T is (4, -4), so the length of ST (which is the height's related? Wait, no, in a parallelogram, area is base × height. Let's find the coordinates:

R(-3, 2), U(-3, -1), S(4, -1), T(4, -4). Wait, no, let's check the coordinates again. Wait, R is at (-3, 2)? Wait, the graph: R is at x=-3, y=2? Wait, the grid: x from -5 to 5, y from -5 to 5. U is at (-3, -1), S is at (4, -1), T is at (4, -4), and R is at (-3, 2)? Wait, no, maybe I misread. Wait, the line from R to S: R is (-3, 2), S is (4, -1)? No, the vertical line from S(4, -1) to T(4, -4) is length 3 (since -1 to -4 is 3 units down). The horizontal distance from U(-3, -1) to S(4, -1) is 4 - (-3) = 7? Wait, no, that can't be. Wait, maybe the base is the length of US, which is from x=-3 to x=4, so 7 units? But the height is the vertical distance? Wait, no, in a parallelogram, the area is base × height, where height is the perpendicular distance between the bases.

Wait, let's find the coordinates properly:

Looking at the graph:

• U is at (-3, -1)
• S is at (4, -1)
• T is at (4, -4)
• R is at (-3, 2)

So, US is a horizontal line from (-3, -1) to (4, -1), so length of US is \( 4 - (-3) = 7 \)? Wait, no, 4 - (-3) is 7? Wait, 4 - (-3) = 7? Yes, because -3 to 4 on the x-axis is 7 units (since 4 - (-3) = 7). Then, the height is the vertical distance between the lines RU and ST. Wait, RU is from (-3, 2) to (-3, -1), which is a vertical line (x=-3) with length 3 (2 - (-1) = 3). Wait, but in a parallelogram, opposite sides are equal and parallel. So RU is vertical (length 3), ST is vertical (from (4, -1) to (4, -4), length 3). US is horizontal (length 7), RT is horizontal? Wait, no, RT would be from (-3, 2) to (4, -4), but no, the sides are RU, US, ST, TR.

Wait, actually, the base can be US, which is horizontal, length 7? No, wait, the vertical distance between the two horizontal sides? Wait, no, RU is vertical, so the height is the horizontal distance? Wait, no, I think I made a mistake. Let's use the formula for the area of a parallelogram: base × height, where base is the length of one side, and height is the perpendicular distance to the opposite side.

Looking at the coordinates:

• U(-3, -1), S(4, -1): so US is horizontal, length \( 4 - (-3) = 7 \)? Wait, no, 4 - (-3) is 7? Wait, 4 - (-3) = 7, yes. Then, the height is the vertical distance from R to the line US. Since US is on y=-1, and R is at (-3, 2), the vertical distance is \( 2 - (-1) = 3 \). Wait, but then area would be 7 × 3 = 21? But that's one of the options. Wait, but let's check again.

Wait, maybe the base is ST? ST is from (4, -1) to (4, -4), length 3. Then the height is the horizontal distance from U to the line ST. ST is on x=4, U is at (-3, -1), so horizontal distance is \( 4 - (-3) = 7 \). Then area is 3 × 7 = 21. Yes, that matches. So the area is 21 square units.

Step1: Determine base and height

Base (length of ST): vertical distance from (4, -1) to (4, -4) is \( |-4 - (-1)| = 3 \).
Height (horizontal distance from U(-3, -1) to ST (x=4)): \( |4 - (-3)| = 7 \).

Step2: Calculate area

Area of parallelogram = base × height = \( 3 × 7 = 21 \).

Answer:

21 square units