In order to determine if the points create a rectangle, first we will label the given points.
A(-2,1), B(4,3), C(8,-3), D(2,-7)
Next we plot them in a coordinate plane and form a quadrilateral.
Examining the figure, we can see that it is not a rectangle. However, we want to prove it mathematically. If this is a rectangle all four angles have to be right angles, which means adjacent sides have to be perpendicular to one another. If we prove that just one of the angles is not a right angle, we will know it is not a rectangle. It looks like ∠ ABD is not a right angle.
Let's try to prove that! It will be enough to show that AB and BD are not perpendicular to each other. To do that, we need to check the product of their slopes. Let's first calculate the slope of these sides starting with AB. To do so, we will substitute (4,3) and (-2,1) into the Slope Formula.
We found that the slope of AB is 13 and that the slope of BD is - 32. Let's calculate their product.
m_(AB) * m_(BD) = - 1/3 * 3/2 = - 3/6 ≠ - 1
Since the product of the slopes of AB and BD is not -1, these segments are not perpendicular. Therefore, the given points cannot be the vertices of a rectangle.
