Question

In $△ABC,AL$, and $CM$ are the perpendiculars from the vertices $A$ and $C$ to $BC$ and $AB$ respectivaly. If $AL$ and $CM$ intersect at $O,$

Prove that:

$\left(i\right)△OMA\sim △OLC$

$\left(ii\right)\frac{OA}{OC}=\frac{OM}{OL}$

Answer 1

$\left(i\right)$

In $△OMA$ and $△OLC,$

$\angle AOM=\angle COL$ [Vertically opposite angles]

$\angle OMA=\angle OLC$ [Both angles are ${90}^{\circ }$]

Hence, by $AA$ criteria of similarity,

$△OMA\sim △OLC$

$\left(ii\right)$

By using the property that the corresponding sides of similar triangle are proportional,

$\frac{OA}{OC}=\frac{OM}{OL}$