Discussion

You wrote:
“Draw the parallelogram with vertices A(8, 5), B(17, 5), C(12, 10), and
D(21, 10) on a coordinate plane. Find the area of the rectangle.”
Here’s the diagram:
Try as I might, I can’t find a rectangle. So I’ll just assume you want the
area of the parallelogram.
The area of a parallelogram is given by A=bh where b is the measure of
the base and h is the measure of the height.

Solution

Since the base of this parallelogram is parallel to the x-axis, the measure of
the base is calculated by the difference in the x-coordinates of the end
points of the base line segment. b=17-8=red%289%29
Note: You could use the distance formula to calculate the measure of the base,
but since the y-coordinates of the two points are equal, the distance formula
degenerates to the square root of the square of the difference in x, or simply
the difference in x.
Also, the height is parallel to the y-axis, so the measure of the height is
calculated by the difference in the y-coordinates of the end points of the
height line segment. One of the end points is given and has a y-coordinate of
10. But the other point is not given. Fortunately, the missing end point
lies on the line segment representing the base. Since that line segment is
parallel to the x-axis, all of the points on that line have the same
y-coordinate. Therefore, the y-coordinate of the missing end point of the
height line segment is equal to the y-coordinate of either end point of the
base line segment, namely 5. Hence, the measure of the height is 10-5=green%285%29
Now we have sufficient information to calculate the area of the parallelogram: A%5Bp%5D=b%2Ah=red%289%29%2Agreen%285%29=blue%2845%29 square units.

Check Answer

Cut off the triangle that was formed on the left of the parallelogram by the
construction of the height line segment. Move this triangle to the left side
of the parallelogram to form a 9 unit by 5 unit rectangle. Multiply 9 times 5
to get 45 which is the area of the rectangle equal to the area of the
parallelogram. Alternatively, you could count the grid squares contained in
the rectangle to verify there are 45 of them.

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