The coordinates of rhombus WXYZ are W(0, 4b), X(2a, 0), Y(0, -4b), and Z(-2a, 0).

Question

Prove: The segments joining the midpoints of a rhombus form a rectangle.

Answer 1

The coordinates of rhombus are W (0, 4b), X (2a, 0), Y(0, -4b) and Z(-2a, 0)

Midpoint formula :

The midpoint of WX

The midpoint of  XY

The midpoint of YZ

The midpoint of ZX

Draw the segments joining the midpoints of the rhombus.

The segments joining the midpoints form a polygon.

Inorder to prove this polygon is rectangle, prove diagonals are congruent.

Find the measure of the diagonals.

The distance between and

Distance formula

The distance betweenand

Distance formula

Since the diagonals have same measures, they congurent.

Comments

Let the points are A=(a, 2b), B=(a, -2b), C=(-a, -2b) and D=(-a, 2b).

To prove the ABCD is a rectangle, prove the opposite sides are equal (AB = CD and AC = BD) and diagonals are equal (AD = BC).

Since AB = CD, AC = BD and AD = BC, ABCD is rectangle.