Question

The diagonals of a rectangle are unequal in length.
(a) True
(b) False

Answer 1

Hint: For solving this problem use the congruence of triangles. We consider a rectangle ‘ABCD’.

Here, we consider two triangles which have two diagonals in each triangle and prove that those two triangles are congruent to each other so that the diagonals are equal.

Complete step by step answer:
Let us consider a rectangle ‘ABCD’ as shown below

Now, let us assume the two triangles $$\mathrm{\Delta }ADC$$ and $$\mathrm{\Delta }BCD$$.
Here we can write three equations
(i)$$DC=CD$$
(ii)$$\mathrm{\angle }ADC=\mathrm{\angle }BCD={90}^0$$, angles in a rectangle are equal to $${90}^0$$.
(iii)$$\mathrm{\angle }DAC=\mathrm{\angle }CBD$$
We know that if two angles and one side of two triangles are equal then we can say that from A.A.S congruence those two triangles are equal.
So, from A.A.S congruence we can write $$\mathrm{\Delta }ADC$$ and $$\mathrm{\Delta }BCD$$ are congruent to each other.
We know that if two triangles are congruent then all the corresponding sides and angles are equal.
So, from the definition of congruence we can say that
$$\Rightarrow AC=BD$$
Hence, we can say that the diagonals of a rectangle are equal.
But we are given that the diagonals of the rectangle are unequal in length. So, the given statement is wrong.

So, the correct answer is “Option b”.

Note: This question is explained in another way.
Considering all the types of quadrilaterals there are only two quadrilaterals which have equal lengths of diagonals, they are squares and rectangles.
Since, all four angles in squares and rectangles are equal to $${90}^0$$ the diagonals will always be equal and cut at midpoints in right angles. This is the definition of squares and rectangles.
So, the given statement is wrong which leaves option (b) as the correct answer.