Question

State whether the following statement is true or false.

in the given figure, $DB\perp BC,DE\perp AB$ and $AC\perp BC$. then $\frac{BE}{DE}=\frac{AC}{BC}$

• A. True
• B. False

Answer 1

The correct option is A True

$\Rightarrow$ $\angle BED=\angle ACB={90}^o$ [ Given ]

Now, according to what is given, $DB\perp BC$ and $AC\perp BC$ we can write,

$\Rightarrow$ $\angle B+\angle C={180}^o$

This clearly means that $BD\parallel CA$

$\Rightarrow$ $\angle EBD=\angle CAB$ [ Alternate angles ]

In $△BED$ and $\angle ACB$

$\Rightarrow$ $\angle BED=\angle ACB={90}^o$

$\Rightarrow$ $\angle EBD=\angle CAB$

$\therefore$ $△BED\sim △ACB$ [ By $AA$ similarity test ]

$\Rightarrow$ $\frac{BE}{AC}=\frac{DE}{BC}$

$\therefore$ $\frac{BE}{DE}=\frac{AC}{BC}$ --- Hence proved