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$AD$.

Answer 1

Step1: Finding the area of$\Delta ABC$

$∆ABC$is right–angled at $A$.

$AB=5cm,BC=13cm$ and $AC=12cm$

now, area of triangle

$\begin{array}{rcl}∆ABC& =& \frac{1}{2}\times altitude\times base\\ & =& \frac{1}{2}\times AB\times AC\\ & =& \frac{1}{2}\times 5cm\times 12cm\\ & =& 30cm²\end{array}$

Hence, area of$∆ABC=30cm²$

Step2:Finding length of $AD$

Area of triangle

$\begin{array}{rcl}A∆ABC& =& \frac{1}{2}\times BC\times AD\\ & \Rightarrow & 30=\frac{1}{2}\times 13\times AD\\ & \Rightarrow & \frac{30\times 2}{13}=AD\\ & \Rightarrow & AD=\frac{60}{13}\\ & \Rightarrow & AD=4.61cm\end{array}$

Therefore the area of$∆ABC=30cm²$ and length of $\begin{array}{rcl}AD& =& 4.61cm\end{array}$