Question

Logically prove the following statement:
Both the diagonals of a rectangle are equal in length.

Answer 1

Hint: Let us prove that the diagonals are equal by proving that the triangles of the rectangle are congruent to each other. In congruent triangles corresponding sides are to be equal.

Complete step-by-step answer:
Let us first draw a rectangle ABCD.

As we can see that the diagonals of the above drawn rectangle are AC and BD.
Now as we know that all angles of the rectangle are equal and equal to \[{90^0}\].
So, \[\angle {\text{A}} = \angle {\text{B}} = \angle {\text{C}} = \angle {\text{D}} = {90^0}\]
And opposite sides of the rectangle are also equal.
So, AD = BC and AB = DC
Now we had to prove that both the diagonals of the rectangle (i.e. AC and BD) are equal.
So, to prove this logically. We had to prove that the \[\Delta {\text{ABC}}\] and \[\Delta {\text{BAD}}\] are congruent.
So, as we know that opposite sides of a rectangle are equal.
So, BC = AD
And all angles are equal to \[{90^0}\].
So, \[\angle {\text{A}} = \angle {\text{B}}\]
And side AB is common to both the triangles \[\Delta {\text{ABC}}\] and \[\Delta {\text{BAD}}\].
So we can say that from the side angle side (SAS) congruence rule of the triangle \[\Delta {\text{ABC}}\] and \[\Delta {\text{BAD}}\] are congruent.
As we know from the congruence rule of triangles that if two triangles are congruent then their corresponding sides must we equal.
So, AC = BD
Hence, the diagonals of the rectangle are equal in length.

Note: Whenever we come up with this type of problem and are asked to prove the result logically then, first we draw the required figure first. And after that we had to prove that the triangles with diagonals as one of their sides are congruent to each other. If they are congruent then from the congruence rule of triangles the diagonals must be equal. This will be an efficient way to prove the result.