Answer: 8 * 8 = (distance between PQ and RS) * 8
distance between PQ and RS = 8" PQRS in units: 16 because 8 (PQ) + 8 (RS)=16, measurement type is units so 16 units
Step-by-step explanation:
*I used A.I to help explain this better.* It should make sense, just read/scan through it, as it explains the question very throughly.
"First, let's find the length of the sides of the rectangle. Since P and Q have the same x-coordinate, we know that PQ is a vertical line segment with length 8 units (since the y-coordinates of P and Q differ by 8). Similarly, since P and Q have the same y-coordinate, we know that RS is a horizontal line segment with length 8 units. Therefore, the length and width of the rectangle are both 8 units.
To find the coordinates of points R and S, we need to consider two cases:
a) Points R and S are to the left of points P and Q.
In this case, we can imagine that the rectangle is reflected across the y-axis, so that points P and Q become points P'(-1, -4) and Q'(-1, 4), respectively. Then, points R and S must lie on the line x=-2 (to the left of point P'), and the distance between them must be 8 units.
Since the area of the rectangle is 64 square centimeters, the length of RS is 8 units, and the length of PQ is 8 units, we know that the distance between PQ and RS (i.e., the height of the rectangle) is also 8 units. This means that the y-coordinates of R and S must differ by 8 units.
Let's choose a y-coordinate for point R. Since R is to the left of P', its x-coordinate is -2, and its y-coordinate must be between -4 and 4 (since the y-coordinates of P' and Q' are -4 and 4, respectively). Let's say that the y-coordinate of R is yR. Then, the y-coordinate of S must be yR + 8.
The area of the rectangle is (length)(width) = (8)(8) = 64 square centimeters. Since PQ is a vertical line segment, its length is the difference between the y-coordinates of P and Q, which is 8 units. Therefore, the length of RS is also 8 units. The distance between PQ and RS (i.e., the height of the rectangle) is also 8 units. Therefore, we can write:
8 * 8 = (distance between PQ and RS) * 8
distance between PQ and RS = 8
So, the y-coordinates of R and S differ by 8 units. Therefore, we can write:
yR + 8 - yR = 8
yR = 0
Therefore, the coordinates of R are (-2, 0), and the coordinates of S are (-2, 8).
b) Points R and S are to the right of points P and Q.
In this case, we can imagine that the rectangle is reflected across the x-axis, so that points P and Q become points P''(1, 4) and Q''(-1, 4), respectively. Then, points R and S must lie on the line y=-6 (to the right of point P''), and the distance between them must be 8 units.
Again, the area of the rectangle is (length)(width) = (8)(8) = 64 square centimeters. Since RS is a horizontal line segment, its length is the difference between the x-coordinates of R and S, which is 8 units. Therefore, the length of PQ is also 8 units. The distance between PQ and RS (i.e., the height of the rectangle) is also 8 units. Therefore, we can write:
8 * 8 = (distance between PQ and RS) * 8
distance between PQ and RS = 8"