Question

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

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

Answer 1

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

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


$Consider\Delta ABDand\Delta ABC,$

$\angle BAD=\angle BAC------\left[commonangle\right]$

$\angle ADB=\angle ABC------\left[{90}^{\circ }\right]$

$ByAAsimilarity,\Delta ADB\sim \Delta ABC$

$Consider\Delta BDCand\Delta ABC,$

$\angle DCB=\angle BCA------\left[commonangle\right]$

$\angle BDC=\angle ABC------\left[{90}^{\circ }\right]$

$ByAAsimilarity,\Delta BDC\sim \Delta ABC$

$Since,\Delta ADB\sim \Delta ABCand\Delta BDC\sim \Delta ABC,\Delta BDC\sim \Delta ADB.$

$\text{Hence all three options are correct.}$