Question

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

• A. $△ADB\sim △ABC$
• B. $△ADB\sim △BDC$
• C. $△ABC\sim △BDC$
• D. None of these

Answer 1

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

The correct options are
A

$△ADB\sim △ABC$

B

$△ADB\sim △BDC$

C

$△ABC\sim △BDC$

Consider $△ABD$ and $△ABC,$
$\angle BAD=\angle BAC$ [common angle]
$\angle ADB=\angle ABC$ [${90}^{\circ }$]
By AA similarity, $△ADB\sim △ABC$
Consider $△BDC$ and $△ABC,$
$\angle DCB=\angle BCA$ [common angle]
$\angle BDC=\angle ABC$ [${90}^{\circ }$]
By AA similarity, $△BDC\sim △ABC$
Since, $△ADB\sim △ABC$ and $△BDC\sim △ABC,$

so, $△BDC\sim △ADB.$
Hence, all three options are correct.