Question

ABC is a right angled at B. D is any point on AB. Seg DE is perpendicular to seg AC IF AD=6cm AB=12cm AC=18cm. Find AE.

Answer 1

Applying Pythogoras Theorem.
$BC62=A{C}^2-A{B}^{2}BC=\sqrt{A{C}^2=A{B}^2}=\sqrt{{18}^2-{12}^2}=\sqrt{180}=6\sqrt{5}cm$
In $\Delta DEA\&\Delta$
$\angle DEA=\angle ABC={90}^{\circ }\left(given\right)\angle A=\angle A\left(common\right)ThisbyAAsimilaritycriterion,\Delta DEA\sim \Delta ABC.\therefore \frac{EA}{BC}=\frac{DA}{AC}.\frac{EA}{6\sqrt{5}}=\frac{6}{18}\therefore EA=\frac{36\sqrt{5}}{18}=2\sqrt{5}cm.$