Question

In the given figure, triangle $ABC$ is right - angled at $B$. Given that $DE=3cm$, $AE=5cm$ and $D$ and $E$ are the mid-points of the sides $AB$ and $AC$ respectively. If $DE\parallel BC$, find the sides $AB$ and $BC$.

• A. $AB=10cm,BC=8cm$
• B. $AB=8cm,BC=10cm$
• C. $AB=8cm,BC=6cm$
• D. $AB=6cm,BC=8cm$

Answer 1

The correct option is C $AB=8cm,BC=6cm$

Given,
$D$ and $E$ are mid-points of $AB$ and $AC$ and $DE\parallel BC$
$\therefore$ By mid point theorem, $DE=\frac{1}{2}BC$
$\Rightarrow BC=2\times DE$
$\Rightarrow BC=2\times 3$
$\therefore BC=6cm$
Since, $E$ is the midpoint of $AC$,
$AC=2\times AE$
$AC=2\times 5$
$AC=10cm$
By pythagoras theorem, $A{B}^2+B{C}^2=A{C}^2$
$\Rightarrow A{B}^2=A{C}^2-B{C}^2$
$\Rightarrow A{B}^2={10}^2-6^2$
$\Rightarrow A{B}^2=100-36=64$
$\therefore AB=8cm$