What is the area of the rectangle formed by the four points (-3, -1), (-1, 3), (1 , 3), (3, 1)?

1 Answer
smendyka
Feb 27, 2018

See a solution process below:

Explanation:

First, we can plot the four points on a coordinate grid:

graph{((x + 3)^2 + (y + 1)^2 - 0.025)((x - 1)^2 + (y - 3)^2 - 0.025)((x - 3)^2 + (y - 1)^2 - 0.025)((x + 1)^2 + (y - 3)^2 - 0.025) = 0 [-10, 10, -5, 5]}

As you can see these points do not form a rectangle. However, if you change the fourth point to #(-1, -3)# this does form a rectangle.

graph{((x + 3)^2 + (y + 1)^2 - 0.025)((x - 1)^2 + (y - 3)^2 - 0.025)((x - 3)^2 + (y - 1)^2 - 0.025)((x + 1)^2 + (y + 3)^2 - 0.025) = 0 [-10, 10, -5, 5]}

The formula for the area of a rectangle is:

#A = l xx w#

We can find the distance between:

#(-3, -1)# and #(1, 3)# to determine the length

#(1, 3)# and #(3, 1)# to determine the width

The formula for calculating the distance between two points is:

#d = sqrt((color(red)(x_2) - color(blue)(x_1))^2 + (color(red)(y_2) - color(blue)(y_1))^2)#

Substituting the values from the points in the problem to find the length gives:

#d_l = sqrt((color(red)(1) - color(blue)(-3))^2 + (color(red)(3) - color(blue)(-1))^2)#

#d_l = sqrt((color(red)(1) + color(blue)(3))^2 + (color(red)(3) + color(blue)(1))^2)#

#d_l = sqrt(4^2 + 4^2)#

#d_l = sqrt(16 + 16)#

#d_l = sqrt(16 xx 2)#

#d_l = sqrt(16)sqrt(2)#

#d_l = 4sqrt(2)#

Substituting the values from the points in the problem to find the width gives:

#d_w = sqrt((color(red)(3) - color(blue)(1))^2 + (color(red)(1) - color(blue)(3))^2)#

#d_w = sqrt(2^2 + (-2)^2)#

#d_w = sqrt(4 + 4)#

#d_w = sqrt(4 xx 2)#

#d_w = sqrt(4)sqrt(2)#

#d_w = 2sqrt(2)#

We can now find the area of the rectangle by multiply the two distances we calculated:

#A = 4sqrt(2) xx 2sqrt(2)#

#A = 4 xx sqrt(2) xx 2 xx sqrt(2)#

#A = 4 xx 2 xx sqrt(2) xx sqrt(2)#

#A = 8 xx sqrt(2)^2#

#A = 8 xx 2#

#A = 16#

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