Determine whether a figure with the given vertices is a rectangle using the Distance Formula.A(4, –7), B(4, –2), C(0, –2), D(0, –7)Question 14 options:Yes; Opposite sides are congruent.Yes; Opposite sides are congruent, and diagonals are congruent.Yes; Opposite sides are parallel.No; Opposite sides are congruent, but diagonals are not congruent.

Accepted Solution

Answer:Yes; Opposite sides are congruent, and diagonals are congruent.Step-by-step explanation:we have[tex]A(4, -7), B(4, -2), C(0, -2), D(0, -7)[/tex]we know thatthe formula to calculate the distance between two points is equal to [tex]d=\sqrt{(y2-y1)^{2}+(x2-x1)^{2}}[/tex] step 1Find the length of the sidesFind the distance ABsubstitute the values[tex]d=\sqrt{(-2+7)^{2}+(4-4)^{2}}[/tex] [tex]d=\sqrt{(5)^{2}+(0)^{2}}[/tex] [tex]AB=5\ units[/tex] Find the distance BCsubstitute the values[tex]d=\sqrt{(-2+2)^{2}+(0-4)^{2}}[/tex] [tex]d=\sqrt{(0)^{2}+(-4)^{2}}[/tex] [tex]BC=4\ units[/tex] Find the distance CDsubstitute the values[tex]d=\sqrt{(-7+2)^{2}+(0-0)^{2}}[/tex] [tex]d=\sqrt{(-5)^{2}+(0)^{2}}[/tex] [tex]CD=5\ units[/tex] Find the distance ADsubstitute the values[tex]d=\sqrt{(-7+7)^{2}+(0-4)^{2}}[/tex] [tex]d=\sqrt{(0)^{2}+(-4)^{2}}[/tex] [tex]AD=4\ units[/tex] Compare the length sidesAB=CDBC=ADthereforeOpposite sides are congruentstep 2Find the length of the diagonalsFind the distance ACsubstitute the values[tex]d=\sqrt{(-2+7)^{2}+(0-4)^{2}}[/tex] [tex]d=\sqrt{(5)^{2}+(-4)^{2}}[/tex] [tex]AC=\sqrt{41}\ units[/tex]Find the distance BDsubstitute the values[tex]d=\sqrt{(-7+2)^{2}+(0-4)^{2}}[/tex] [tex]d=\sqrt{(-5)^{2}+(-4)^{2}}[/tex] [tex]BD=\sqrt{41}\ units[/tex]Compare the length of the diagonalsAC=BDthereforeDiagonals are congruentThe figure is a rectangle, because Opposite sides are congruent, and diagonals are congruent