Question

Identify the number of triangles similar to $△$ABC if B is a right angle, and BD is perpendicular to AC and DE is perpendicular to BC.

• A. 2
• B. 3
• C. 4
• D. 6

Answer 1

The correct option is C

4

BD is the perpendicular drawn to the hypotenuse.
In $△ABD$ and $△BDC$
$\angle ADB=\angle BDC={90}^{\circ }$
$\angle A=\angle DBC$ [Since, in $△BDC,\angle DBC={90}^{\circ }-\angle C=\angle A$ and in $△ABC,\angle A={90}^{\circ }-\angle C$]

Therefore, $△ADB\sim \Delta BDC$ [by AA similarity]
Similarly, in $△BDC,$ DE is the perpendicular drawn to the hypotenuse BC.
In $△BDE$ and $△BDC$
$\angle BED=\angle BDC={90}^{\circ }$
$\angle DBC=\angle DBC$ [Common Angle]
Therefore, $△BED\sim △BDC$ [by AA similarity]
Hence, $△ADB\sim \Delta BDC\sim \Delta BED\sim \Delta DEC\sim \Delta ABC.$
Therefore, there are 4 triangles similar to $△ABC.$