In a right-angled triangle show that the hypotenuse is the longest side.
Answer:
- Let ^@PQR^@ be a right-angled triangle such that ^@\angle Q = 90^ \circ.^@
- As the sum of angles of a triangle is ^@180^ \circ^@.
Now, in ^@\triangle PQR,^@ we have @^ \begin{aligned} & \angle P + \angle Q + \angle R = 180 ^ \circ \\ \implies & \angle P + 90 ^ \circ + \angle R = 180 ^ \circ && [As, \angle Q = 90^\circ] \\ \implies & \angle P + \angle R = 180 ^ \circ - 90 ^ \circ \\ \implies & \angle P + \angle R = 90 ^ \circ \\ \end{aligned} \\@^ As the measure of ^@\angle Q ^@ is equal to the sum of measures of ^@\angle P ^@ and ^@\angle R ^@, we have @^ \begin{aligned} & \angle Q > \angle P \\ \implies & PR > QR & \ldots \text { (1) } && [\text {Side opposite to greater angle is greater.}] \\ \end{aligned}@^ Also, @^ \begin{aligned} & \angle Q > \angle R \\ \implies & PR > PQ & \ldots \text { (2) } && [\text{Side opposite to greater angle is greater.}] \end{aligned}@^ - By equation ^@ \text { (1) } ^@ and ^@ \text { (2) } ^@, we have @^ PR > QR \text { and } PR > PQ \\ @^ As ^@ PR ^@ is greater than both the sides, ^@ \bf PR \bf ^@ is longest side.
