Question

In $△ABC,\angle A$ is a right angle. If $AD\perp BC$ and $DϵBC$ then which of the following statements are true?
1. $A{D}^2=BD\cdot DC$
2. $B{C}^2=AD\cdot DC$
3. $A{B}^2=BD\cdot BC$
4. $A{C}^2=CD\cdot BD$.

• A. Statements $1$ and $4$ are true
• B. Statements $1$ and $2$ are true
• C. Statements $1,3$ and $4$ are true
• D. Statements $1$ and $3$ are true

Answer 1

Answer: (D)

Statements $1$ and $3$ are true

In $△ABC,\angle A$ is a right angle.
$\overline{AD}\perp \overline{BC}$.
$B-D-C$
$△ABC\sim △DBA(\because AA).....\left(1\right)$
$\therefore \frac{AB}{BD}=\frac{BC}{AB}$
$\therefore A{B}^2=BD\cdot BD$
$△ABC\sim △DAC(\because AA)....\left(2\right)$
$\frac{BC}{AC}=\frac{AC}{CD}$
$A{C}^2=BC\cdot CD$
From result $\left(1\right)$ and $\left(2\right)$,
$△DBA\sim △DAC$
$\therefore \frac{BD}{AD}=\frac{AD}{CD}$
$\therefore A{D}^2=BD\cdot CD$
Therefore, statements $\left(1\right)$ and $\left(3\right)$ are true.