# In an equilateral triangle $A B C$ if $A D perp B C$, then $A D^{2}=$

Question:

In an equilateral triangle $A B C$ if $A D \perp B C$, then $A D^{2}=$

(a) $\mathrm{CD}^{2}$

(b) $2 \mathrm{CD}^{2}$

(c) $3 \mathrm{CD}^{2}$

(d) $4 \mathrm{CD}^{2}$

Solution:

In an equilateral $\triangle \mathrm{ABC}, \mathrm{AD} \perp \mathrm{BC}$.

In ΔADC, applying Pythagoras theorem, we get

$\mathrm{AC}^{2}=\mathrm{AD}^{2}+\mathrm{DC}^{2}$

$\mathrm{BC}^{2}=\mathrm{AD}^{2}+\mathrm{DC}^{2}(\because \mathrm{AC}=\mathrm{BC})$

$(2 \mathrm{DC})^{2}=\mathrm{AD}^{2}+\mathrm{DC}^{2}(\because \mathrm{BC}=2 \mathrm{DC})$

$4 D C^{2}=A D^{2}+D C^{2}$

$3 \mathrm{DC}^{2}=\mathrm{AD}^{2}$

$3 \mathrm{CD}^{2}=\mathrm{AD}^{2}$

Hence, the correct option is (c).