Question

State true or false.
In an equilateral triangle, a perpendicular drawn from one of the vertices to the opposite side bisects the side.

• A. True
• B. False

Answer 1

The correct option is A True


In $△$ABC,
$\angle A=\angle B=\angle C={60}^{\circ }$ since it is equilateral.
Consider $△$ABD,
$\angle$ADB = ${90}^{\circ }$
$\angle$ABC = ${60}^{\circ }$
$\Rightarrow \angle BAD={180}^{\circ }-{60}^{\circ }-{90}^{\circ }\Rightarrow \angle BAD={30}^{\circ }$
Similarly, $\angle CAD={30}^{\circ }$
Consider $△$ADB and $△$ADC,
AB = AC = a (All sides are equal)
AD = AD (Common side)
$\angle BAD=\angle CAD={30}^{\circ }$
Now $△ABD\cong △ACD$
(according to SAS congruency)
Therefore, BD = DC.