Place one vertex at the origin and one side along the x-axis.
See solution.
Practice makes perfect
We are asked to prove that if the diagonals of a parallelogram are congruent, then it is a rectangle.
We are asked to write a coordinate proof, so let's place the parallelogram in a coordinate plane so that one vertex is at the origin and one side is on the x-axis.
We can use the Distance Formula to express the lengths of the diagonals using the coordinates of the endpoints.
The distance
between points(x_1,y_1)and(x_2,y_2)is
sqrt((x_2-x_1)^2+(y_2-y_1)^2).Let's write and simplify an expression for the length of AC first.
This can only be true if b=0 or d=0.
Since D(d,0) is not at the origin, d≠0.
This means that the x-coordinate of B(b,c) is 0, so B is on the y-axis.
Since the coordinate axes are perpendicular and the sides of the parallelogram are parallel to the coordinate axes, we can see that the parallelogram has four right angles.
By definition, this means that the parallelogram is a rectangle.
