Question

Show that in a right-angled triangle, the hypotenuse is the longest side.

Answer 1

Let us consider a right angle triangle $∆\mathrm{ABC}$, right angled at $\mathrm{B}$.

To prove:

Hypotenuse is the longest side in $∆\mathrm{ABC}$

Proof:

Step 1: In $∆\mathrm{ABC}$, $\angle \mathrm{B}={90}^0$

Step 2: In $∆\mathrm{ABC}$, $\angle \mathrm{A}$ and $\angle \mathrm{C}$ are acute angles. So, $\angle \mathrm{A}$$<{90}^0$ and $\angle \mathrm{B}$$<{90}^0$

Step 3: Based on the above two steps it is clear that $\angle \mathrm{B}={90}^0$ is greatest among all the angles of $∆\mathrm{ABC}$..

Theorem: Side opposite to the greater angle of any triangle is the longest side.

Taking reference of the above theorem we can conclude that hypotenuse $\mathrm{AC}$ is the longest side in $∆\mathrm{ABC}$.

Thus, hypotenuse is the longest side in right angled triangle.