Question

Prove that the diagonals of a rhombus intersect at right angles. (A rhombus is a quadrilateral with sides of equal lengths.)

Short answer

Yes, the diagonals of a rhombus do intersect at right angles.

Step-by-step solution

Draw the rhombus and its diagonals

First, a rhombus is drawn labeling its vertices as A, B, C, and D. Then, the diagonals of the rhombus are drawn, which intersect each other at a point, labeled as E.

Prove the two triangles formed by the diagonals are congruent

By drawing the diagonals, two triangles are formed, namely triangle AED and triangle BEC. In a rhombus, opposite angles are equal, so angle AED = angle BEC and angle DEA = angle ECB. Furthermore, side DE is equal to side BE, as they represent the same line segment. Therefore, by SAS postulate (Side-Angle-Side), triangle AED is congruent to triangle BEC (∆AED ≅ ∆BEC)

Calculate the angles formed by the intersecting diagonals

Since the two triangles are congruent, their corresponding angles are equal. This means angle DEA = angle BEC and angle AED = angle ECB. But DEA and BEC are alternate angles, thus, line AB is parallel to line DC. In a similar manner, it can also be deduced that line BC is parallel to line AD. So, the diagonals are actually bisecting the angles of the rhombus.

Prove the diagonals meet at right angles

Because the diagonals bisect the angles of the rhombus, this means that they divide the angles into two equal parts. Hence, angle AEB and angle DEB are both half of angle AED. Because AED = BEC, each of these angles equals half of this value. But we know the entire angle AED or BEC is a straight line, equating to 180 degrees. So, each of angle AEB and angle DEB equals half of 180 degrees, which is 90 degrees. Therefore, diagonals of a rhombus intersect each other at right angles.