Question

In a $\Delta ABC$, BD and CE are the altitudes. Prove that $\Delta ADB$ and $\Delta AEC$ are similar. Is $\Delta CDB\sim \Delta BEC$ ? [4 MARKS]

Answer 1

Concept: 1 Mark
Application: 3 Marks

In $\Delta ABD$ and $\Delta AEC$, we have

$\angle ADB=\angle AEC$ [Each equal to ${90}^{\circ }$]

$\angle BAD=\angle EAC$

So, by AA-criterion of similarity, we have

$\Delta BDA\sim \Delta CEA$ or, $\Delta ADB\sim \Delta AEC$.

Clearly, $\Delta CDB$ is not similar to $\Delta BEC$, because they are not equiangular.