Question

Are the diagonals of a rectangle equal? Why?

Answer 1

Verify the given statement:

Consider the triangles $△ADC$ and $△BDC$

All angles of a rectangle are right angles. Hence, $\angle ADC=\angle BCD=90°$

In right angled triangle $△ADC$,by Pythagoras' theorem

$A{D}^2+D{C}^2=A{C}^2$ $...\left(i\right)$

In right angled triangle ,$△BDC$ by Pythagoras' theorem

$B{C}^2+D{C}^2=B{D}^2$ $...\left(ii\right)$

The opposite sides of a rectangle are equal. Hence,

$AD=BC$$...\left(iii\right)$

From $\left(i\right),\left(ii\right)$ and $\left(iii\right)$ we get that

$A{C}^2=B{D}^2$

$\Rightarrow$$AC=BD$

Thus diagonals of a rectangle are equal.

In a rectangle all the angles are equal and therefore the diagonals are equal.

Hence,it is proved that the diagonals of the rectangle are equal.