Question

In a right triangle $\Delta ABC$ with right angle at B, BD is a perpendicular drawn to the hypotenuse AC. Which of the following statements are true?

• A. $\Delta ADB$ ~ $\Delta ABC$
• B. $\Delta ADB$ ~ $\Delta BDC$
• C. $\Delta ABC$ ~ $\Delta BDC$
• D. None of these

Answer 1

Answer: (A), (B), (C)

The correct options are
A $\Delta ADB$ ~ $\Delta ABC$
B $\Delta ADB$ ~ $\Delta BDC$
C $\Delta ABC$ ~ $\Delta BDC$


Consider $\Delta ABD$ and $\Delta ABC$,
$\angle BAD=\angle BAC$ - - - - - - [common angle]
$\angle ADB=\angle ABC$ - - - - - - [${90}^{\circ }$]
By AA similarity, $\Delta ADB\sim \Delta ABC$
Consider $\Delta BDC\text{and}\Delta ABC$,
$\angle DCB=\angle BCA$ - - - - - - [common angle]
$\angle BDC=\angle ABC$ - - - - - -[90${}^{\circ }$]
By AA similarity, $\Delta BDC\sim \Delta ABC$
Since, $\Delta ADB\sim \Delta ABC$ and $\Delta BDC\sim \Delta ABC$, $\Delta BDC\sim \Delta ADB$.
Hence all three options are correct.