Question

In $△ABC,AD$ and $BE$ are altitudes $\frac{ar\left(△DEC\right)}{ar\left(△ABC\right)}=\frac{D{C}^3}{A{C}^3}$.

State whether statement is true or false.

• A. True
• B. False

Answer 1

The correct option is B False

$AD$ and $BE$ are altitudes.

$\angle AEA=\angle BDA$ and $\angle DAC=\angle DBC$.

Hence, quadrilateral $ABDE$ is a cyclic quad.

Hence opposite angles are supplementary,

$\angle AED=180-\angle B$

$\Rightarrow \angle DEC=\angle B$

$\Rightarrow in△DEC,\angle EDC=\angle A$

Hence, the two triangles $ABC$ and $DEC$ are similar.

Thus the areas of the triangles are proportional to the squares of corresponding sides.

$\Rightarrow \frac{Ar\left(DEC\right)}{Ar\left(ABC\right)}=\frac{D{C}^2}{A{C}^2}=\frac{D{E}^2}{A{B}^2}$