In a right-angled triangle show that the hypotenuse is the longe | Math Question and Answer

Answer:

Step by Step Explanation:

Let $P Q R$ be a right-angled triangle such that $∠ Q = 90^{\circ} .$
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.

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