Question

# $\Delta ABC$ is right-angled at $A$.$AD$ is perpendicular to $BC$, $AB=5cm,BC=13cm$and $AC=12cm$. Find the area of$\Delta ABC$. Also, find the length of$AD$.

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Solution

## Step1: Finding the area of$\Delta ABC$ $∆ABC$is right–angled at $A$.$AB=5cm,BC=13cm$ and $AC=12cm$now, area of triangle$\begin{array}{rcl}∆ABC& =& \frac{1}{2}×altitude×base\\ & =& \frac{1}{2}×AB×AC\\ & =& \frac{1}{2}×5cm×12cm\\ & =& 30cm²\end{array}$Hence, area of$∆ABC=30cm²$Step2:Finding length of $AD$Area of triangle$\begin{array}{rcl}A∆ABC& =& \frac{1}{2}×BC×AD\\ & ⇒& 30=\frac{1}{2}×13×AD\\ & ⇒& \frac{30×2}{13}=AD\\ & ⇒& AD=\frac{60}{13}\\ & ⇒& AD=4.61cm\end{array}$Therefore the area of$∆ABC=30cm²$ and length of $\begin{array}{rcl}AD& =& 4.61cm\end{array}$

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