Question

Identify the number of triangles similar to $\Delta ABC$ 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 $\Delta ABD$ and $\Delta BDC$
$\angle ADB=\angle BDC={90}^{\circ }$
$\angle A=\angle DBC$ (Since in $\Delta BDC,\angle DBC={90}^{\circ }-\angle C=\angle A$
and in $\Delta ABC\angle A={90}^{\circ }-\angle C)$
Therefore,$\Delta ADB\sim \Delta BDC$ by AA similarity
Similarly, in $\Delta BDC$, DE is the perpendicular drawn to the hypotenuse BC.
In $\Delta BDE$ and $\Delta BDC$
$\angle BED=\angle BDC={90}^{\circ }$
$\angle DBC=\angle DBC$ (Common Angle)
Therefore, $\Delta BED\sim \Delta BDC$ by AA similarity
Hence, $\Delta ADB\sim \Delta BDC\sim \Delta BED\sim \Delta DEC\sim \Delta ABC$.
Therefore, there are 4 triangles similar to $\Delta ABC$.