Question

In the adjoining figure, $ AE = DC = 13 cm BE = 5 cm, \angle ABC=90°\& AD = EC = x cm$. Calculate the length of $ AB$ & the value of $ x$.

Answer 1

Step 1: Analyze $△ABE$

In triangle$△ABE$,

Given: $AE=13cm,BE=5cm,\angle ABE=90°$

$\therefore$$A{E}^2=A{B}^2+B{E}^2$

$\Rightarrow$$169=A{B}^2+25$

$\Rightarrow$$A{B}^2=169–25$

$\Rightarrow$ $A{B}^2=144$

$\Rightarrow$ $AB=12cm$

Step 2: Analyze $△DBC$

In $△DBC$

Given, $DC=13cm,BE=5cm,\angle DBC=90°$

and, $AD=EC=xcm$

From the figure, we can see that

$DB=AB-x$ and $BC=BE+x$

Step 3: Calculate $x$

$\therefore$$D{C}^2=D{B}^2+B{C}^2$

$\Rightarrow$${13}^2={\left(AB-x\right)}^2+{\left(BE+x\right)}^2$

$\Rightarrow$$169={\left(12-x\right)}^2+{\left(5+x\right)}^2$

$\Rightarrow$$169=144-24x+x^2+25+10x+x^2$

$\Rightarrow$$169=169-14x+2x^2$

$\Rightarrow$$14x=2x^2$

$\Rightarrow$ $x=\frac{\left(2x\right)7}{\left(2x\right)}$

$\Rightarrow$ $x=7cm$

Hence, the value of $x$ is $7cm$.