Question

Let $D$ be the middle point of the side $BC$ of a triangle $ABC$. If the triangle $ADC$ is equilateral, then $a^2:b^2:c^2$ is equal to

• A. $1:4:3$
• B. $4:1:3$
• C. $4:3:1$
• D. $3:4:1$

Answer 1

Answer: (B)

$4:1:3$

$D=$ mid point $\Rightarrow BD=DC$

equilateral $△\Rightarrow AC=DC=AD$

$\alpha =\frac{180-120}{2}=30$

$\frac{a}{\sin A}=\frac{b}{\sin B}=\frac{c}{\sin C}\left(sincelaw\right)$

$\Rightarrow \frac{a}{\sin ({60}^o+{30}^o)}=\frac{b}{\sin {30}^o}$$=\frac{c}{\sin {60}^o}$

$\Rightarrow \frac{a}{1}=\frac{b}{1/2}=\frac{c}{\sqrt{3}/2}=k$

$a=k,b=\frac{k}{2}c=\frac{\sqrt{3}}{2}k$

$a^2:b^2:c^2=1:\frac{1}{4}:\frac{3}{4}$

$=4:1:3$

$\therefore$ Option $B$