Question

If the square on one side of a triangle is equal to the sum of the squares on the other two sides, then those two sides contain ____

Answer 1

Let $AD$ be drawn from point $A$ at a right angle to $C$.

Let $AD$ be made equal to $AB$.

Join $DC$

Since $DA$ is equal to $AB$, $D{A}^2=A{B}^2$

Add $A{C}^2$ on both the sides,

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

Also, $D{C}^2=D{A}^2+A{C}^2$ ....Since $\angle DAC$ is a right angle

Since $B{C}^2=A{B}^2+A{C}^2$, then $B{C}^{2}D{C}^2$

Therefore, side $DC$ $=$ side $BC$

Now compare triangle $DAC$ to triangle $BAC$

Side $DA$ was constructed to be equal to $AB$, side $AC$ is common, and the two bases $DC$ and $BC$ were shown to be equal.

Therefore, $△DAC$ $=$ $△BAC$

Thus, $\angle DAC=\angle BAC$

Since, $\angle DAC$ is a right angle, therefore $\angle BAC$ is a right angle.