Question

ABC is an equilateral triangle of side 2a. Find each of its altitudes.

• A. $\sqrt{2}a$
• B. $\sqrt{3a}$
• C. $\sqrt{3}a$
• D. $\sqrt{2a}$

Answer 1

The correct option is C $\sqrt{3}a$


It is given that ABC is an equilateral triangle of side 2a and $AD\perp BC$.
In $\Delta ADB$ and $\Delta ADC$, we have,
AB = AC [Given]
AD = AD [Given]
$\angle ADB=\angle ADC$ [equal to ${90}^{\circ }$]

Therefore, $\Delta ADB\cong \Delta ADC$ by RHS congruence.
Hence, BD = DC [by CPCT]

In right angled $\Delta ADB$,
$A{B}^2=A{D}^2+B{D}^2$
$(2a{)}^2=A{D}^2+a^2$
$\Rightarrow A{D}^2=4a^2-a^2$
$\Rightarrow A{D}^2=3a^2$
$\Rightarrow AD=\sqrt{3}a$