Question

the formula for the area of a rectangle is ( a = lw ). what is the area of rectangle wxyz? 16 square units 18 square units 20 square units 22 square units

Explanation:

Step1: Find coordinates of vertices

First, identify the coordinates of each vertex:
- \( W(-3, 1) \)
- \( X(3, 3) \)
- \( Y(4, 0) \)
- \( Z(-2, -2) \)

Step2: Calculate length and width

To find the length and width of the rectangle, we can use the distance formula or count the horizontal and vertical distances between appropriate points. Alternatively, we can use the grid to find the horizontal and vertical spans.

Looking at the horizontal distance (length) between \( W(-3, 1) \) and \( Y(4, 0) \): The horizontal change is \( 4 - (-3) = 7 \)? Wait, no, maybe better to use the vertical and horizontal differences between two adjacent vertices. Wait, actually, for a rectangle, the length and width can be found by the distance between two points in the x-direction and y-direction.

Wait, maybe a better approach: Use the coordinates to find the length and width. Let's find the distance between \( W(-3, 1) \) and \( X(3, 3) \) for one side, and between \( X(3, 3) \) and \( Y(4, 0) \) for the adjacent side? No, that's not right. Wait, actually, in a rectangle, opposite sides are equal and adjacent sides are perpendicular.

Alternatively, use the grid to count the number of units for length and width. Let's find the horizontal distance (length) and vertical distance (width).

Looking at the x-coordinates of \( W(-3, 1) \) and \( Y(4, 0) \): The horizontal distance is \( 4 - (-3) = 7 \)? No, that's not correct. Wait, maybe I made a mistake. Let's check the coordinates again. Wait, maybe the rectangle is not aligned with the axes, but we can use the formula for the area of a rectangle by finding the length and width using the distance between two points.

Wait, another method: The area of a rectangle can also be calculated using the formula \( A = |(x_2 - x_1)(y_3 - y_1) - (x_3 - x_1)(y_2 - y_1)| \) for a quadrilateral with vertices \( (x_1,y_1), (x_2,y_2), (x_3,y_3), (x_4,y_4) \) in order. But maybe easier to use the grid.

Wait, let's list the coordinates again:
- \( W(-3, 1) \)
- \( X(3, 3) \)
- \( Y(4, 0) \)
- \( Z(-2, -2) \)

Let's find the vector from \( W \) to \( X \): \( (3 - (-3), 3 - 1) = (6, 2) \)
Vector from \( X \) to \( Y \): \( (4 - 3, 0 - 3) = (1, -3) \)
Wait, that's not perpendicular. Maybe I messed up the vertices. Wait, maybe the correct vertices are \( W(-3, 1) \), \( X(3, 3) \), \( Y(4, 0) \), \( Z(-2, -2) \). Wait, maybe the rectangle is actually a parallelogram, but the problem says it's a rectangle. Wait, maybe the coordinates are:

Wait, looking at the grid:
- \( W \) is at (-3, 1)
- \( X \) is at (3, 3)
- \( Y \) is at (4, 0)
- \( Z \) is at (-2, -2)

Wait, maybe the length is the distance between \( W(-3, 1) \) and \( Y(4, 0) \): No, that's not a side. Wait, maybe the correct way is to find the length and width by counting the number of units between two points. Let's find the horizontal distance (length) between \( W(-3, 1) \) and \( X(3, 3) \): The horizontal change is \( 3 - (-3) = 6 \), vertical change is \( 3 - 1 = 2 \). Then the distance is \( \sqrt{6^2 + 2^2} = \sqrt{40} \), which is not helpful. Wait, maybe I made a mistake in the coordinates.

Wait, maybe the coordinates are:
- \( W(-3, 1) \)
- \( X(3, 3) \)
- \( Y(4, 0) \)
- \( Z(-2, -2) \)

Wait, another approach: Use the formula for the area of a rectangle given by the coordinates of the vertices. The area can be calculated using the shoelace formula.

Shoelace formula: For vertices \( (x_1,y_1), (x_2,y_2), (x_3,y_3), (x_4,y_4) \), the area is \( \frac{1}{2} |x_1y_2 + x_2y_3 + x_3y_4 + x_4y_1 - (y_1x_2 + y_2x_3 + y_3x_4 + y_4x_1)| \)

Plugging in the coordinates:
\( x_1 = -3, y_1 = 1 \)
\( x_2 = 3, y_2 = 3 \)
\( x_3 = 4, y_3 = 0 \)
\( x_4 = -2, y_4 = -2 \)

Calculate the first part: \( x_1y_2 + x_2y_3 + x_3y_4 + x_4y_1 \)
\( (-3)(3) + (3)(0) + (4)(-2) + (-2)(1) = -9 + 0 - 8 - 2 = -19 \)

Calculate the second part: \( y_1x_2 + y_2x_3 + y_3x_4 + y_4x_1 \)
\( (1)(3) + (3)(4) + (0)(-2) + (-2)(-3) = 3 + 12 + 0 + 6 = 21 \)

Then the area is \( \frac{1}{2} | -19 - 21 | = \frac{1}{2} | -40 | = \frac{1}{2} \times 40 = 20 \)? Wait, but that's not matching. Wait, maybe I used the wrong vertices. Wait, maybe the rectangle is WXYZ with vertices W(-3,1), X(3,3), Y(4,0), Z(-2,-2). Wait, maybe the shoelace formula was applied incorrectly. Let's recheck:

First part: \( x_1y_2 = (-3)(3) = -9 \)
\( x_2y_3 = (3)(0) = 0 \)
\( x_3y_4 = (4)(-2) = -8 \)
\( x_4y_1 = (-2)(1) = -2 \)
Sum: \( -9 + 0 - 8 - 2 = -19 \)

Second part: \( y_1x_2 = (1)(3) = 3 \)
\( y_2x_3 = (3)(4) = 12 \)
\( y_3x_4 = (0)(-2) = 0 \)
\( y_4x_1 = (-2)(-3) = 6 \)
Sum: \( 3 + 12 + 0 + 6 = 21 \)

Difference: \( -19 - 21 = -40 \), absolute value 40, half of that is 20. Wait, but the options include 20 square units. So that's the answer.

Wait, but maybe a simpler way: Count the number of units for length and width. Let's find the horizontal distance between W(-3,1) and Y(4,0): The horizontal change is 4 - (-3) = 7, vertical change is 0 - 1 = -1. No, that's not helpful. Alternatively, look at the vertical distance between X(3,3) and Y(4,0): vertical change is 0 - 3 = -3, horizontal change is 4 - 3 = 1. No. Wait, maybe the length is 5 and width is 4? No, 5*4=20. Ah, maybe the length is 5 and width is 4? Wait, let's check the coordinates again.

Wait, another approach: The rectangle has length and width such that when multiplied, give 20. The options are 16, 18, 20, 22. So 20 is an option. Using the shoelace formula, we got 20. So that's the area.

Step3: Confirm with area formula

The formula for the area of a rectangle is \( A = lw \). If we found the length and width correctly, multiplying them should give 20. From the shoelace formula, we got 20, which matches one of the options.

Answer:

20 square units (Option: 20 square units)