Question

prove that the ratio of diagonal to side in a square=√2.

Answer 1

Let's consider Quadrilateral ABCD with diagonal AC



Let the side of the square be a............(1)

In ∆ADC

Angle D = 90°. ........angles of a square

By Pythagoras theorem


${ad}^{2} + {dc}^{2} = {ac}^{2}$

From (1)


${a}^{2} + {a}^{2} = {ac}^{2}$


$2 {a}^{2} = {ac}^{2}$
$2 {side}^{2} = {diagonal}^{2}$


$\frac{ {diagonal}^{2} }{ {side}^{2} } = 2$


$\frac{diagonal}{side} = \sqrt{2}$