Question

The art club is designing a rectangular mural for the school hallway. Three corners are located in a coordinate plane at the following locations: (–1, –1), (–1, 1), and (4, 1).

Overdue test question, If anyone know or can figure this out, feel free to help. Thank you.

Answer 1

To find the fourth corner of the rectangular mural, we can use the fact that opposite sides of a rectangle are parallel and perpendicular. This means that if we draw a line between the first two points, we can find the direction of one side of the rectangle, and if we draw a line between the second and third points, we can find the direction of the adjacent side of the rectangle. The intersection of these two lines will give us the fourth corner of the rectangle.

First, let's find the direction of the side of the rectangle that connects the first two points:

Slope of line connecting (–1, –1) and (–1, 1) = (change in y) / (change in x) = (1 - (-1)) / (-1 - (-1)) = 2 / (-2) = -1

So the side of the rectangle that connects the first two points has a slope of -1. We also know that this line passes through the midpoint of the segment connecting these two points, which is ((-1 + (-1))/2, (-1 + 1)/2) = (-1, 0).

Using point-slope form, we can write the equation of this line as:

y - 0 = -1(x - (-1))

y = -x - 1

Next, let's find the direction of the side of the rectangle that connects the second and third points:

Slope of line connecting (–1, 1) and (4, 1) = (change in y) / (change in x) = (1 - 1) / (4 - (-1)) = 0 / 5 = 0

So the side of the rectangle that connects the second and third points has a slope of 0. We also know that this line passes through the midpoint of the segment connecting these two points, which is ((-1 + 4)/2, (1 + 1)/2) = (1.5, 1).

Using point-slope form, we can write the equation of this line as:

y - 1 = 0(x - 1.5)

y = 1

Now we have two equations for the sides of the rectangle:

y = -x - 1 (from the first two points)

y = 1 (from the second and third points)

To find the fourth corner of the rectangle, we need to find the point where these two lines intersect. We can do this by setting the two equations equal to each other:

-x - 1 = 1

-x = 2

x = -2

Now that we know that x = -2, we can substitute this value into either equation to find the corresponding value of y:

y = -(-2) - 1 = 1

Therefore, the fourth corner of the rectangular mural is located at (-2, 1) in the coordinate plane.