Question

Which statement explains how you could use coordinate geometry to prove that quadrilateral ABCD is a square?

Shape ABCD is shown. Point A is at negative 3, 5. Point B is at 1, 7. Point C is at 3, 3. Point D is at negative 1, 1.

Answer 1

Options are not shown but…
A square has 4 equal length sides and a right angle

Or two equal length diagonals which are perpendicular

Or 4 right angled corners and two adjacent sides equal

Or perpendicular diagonals with the same midpoint

Using the last one…
Gradient of AC = (5 - 3)/(-3 - 3) = 2/-6 = -1/3
Gradient of BD = (7 - 1)/(1 - -1) = 6/2 = 3
-1/3 x 3 = -1 so AC and BD are perpendicular
Midpoint of AC = ((-3 + 3)/2 , (5 + 3)/2) = (0, 4)
Midpoint of BD = ((1 + -1)/2 , (7 + 1)/2) = (0, 4)
QED