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 $\Delta ABC$ ~ $\Delta BDC$


$Consider\Delta ABDand\Delta ABC,\angle BAD=\angle BAC------\left[commonangle\right]\angle ADB=\angle ABC------\left[{90}^{\circ }\right]ByAAsimilarity,\Delta ADB\sim \Delta ABCConsider\Delta BDCand\Delta ABC,\angle DCB=\angle BCA------\left[commonangle\right]\angle BDC=\angle ABC------\left[{90}^{\circ }\right]ByAAsimilarity,\Delta BDC\sim \Delta ABCSince,\Delta ADB\sim \Delta ABCand\Delta BDC\sim \Delta ABC,\Delta BDC\sim \Delta ADB.\text{Hence all three options are correct.}$