Question

Four points are drawn on the coordinate plane and connected with straight lines to form a rectangle. Three of the vertices of the rectangle are located at (2, 1), (2, 4), and (-4, 4).

What are the coordinates of the fourth vertex of the rectangle?
What are the dimensions of the rectangle?
What is the area of the rectangle?

Answer 1

Answer:

The area of the rectangle is 18 square units.

Step-by-step explanation:

To find the coordinates of the fourth vertex of the rectangle, we can use the fact that opposite sides of a rectangle are parallel and equal in length.

The segment connecting (2, 1) and (2, 4) is a vertical line, so the fourth vertex must also lie on a vertical line passing through (-4, 4).

Similarly, the segment connecting (2, 1) and (-4, 4) is a diagonal line with slope (4-1)/(-4-2) = 3/-6 = -1/2. The opposite side of the rectangle is parallel to this line and passes through (2, 4). Therefore, the fourth vertex must lie on a line with slope -1/2 passing through (2, 4).

The intersection of these two lines is the fourth vertex. To find the coordinates, we can set the equations of the two lines equal to each other:

(x - (-4)) / (y - 4) = 0 (vertical line passing through (-4,4))

(x - 2) / (y - 4) = -1/2 (line with slope -1/2 passing through (2,4))

Solving for x and y, we get:

x = -4 (from the first equation)

y = 2 (substituting x=-4 into the second equation)

Therefore, the coordinates of the fourth vertex are (-4, 2).

To find the dimensions of the rectangle, we can calculate the lengths of the sides. The side connecting (2, 1) and (2, 4) has length 4-1=3, and the side connecting (2, 4) and (-4, 4) has length 2-(-4)=6. Therefore, the dimensions of the rectangle are 3 and 6.

To find the area of the rectangle, we can multiply the length and width. Therefore, the area of the rectangle is:

Area = length x width = 3 x 6 = 18.

So, the area of the rectangle is 18 square units.

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