Question

Area of $\Delta BRC:$ Area of the $\Delta ABC$ is

• A. $4:3$
• B. $4:5$
• C. $3:7$
• D. $4:7$

Answer 1

The correct option is D $4:7$

Vector area of $\Delta BRC=\frac{1}{2}\{\overrightarrow{BR}\times \overrightarrow{BC}\}$

$=\frac{1}{2}\left\{\right(\frac{1}{7}(2\overline{b}+\overline{c})-\overline{b})\times (\overline{c}-b\left)\right\}$

$=\frac{1}{14}(\overline{c}-5\overline{b})\times (\overline{c}-\overline{b})$

$=\frac{1}{14}\{-4(\overline{b}\times \overline{c}\left)\right\}$

Area of $\Delta BRC=\frac{2}{7}|\overline{b}\times \overline{c}|$

Vector Area of $\Delta ABC=\frac{1}{2}\overrightarrow{BA}\times \overrightarrow{BC}$

$=\frac{1}{2}\{-\overline{b}\times (\overline{c}-\overline{b}\left)\right\}$

$=-\frac{1}{2}(\overline{b}\times \overline{c})$

Area of $\Delta ABC=\frac{1}{2}|\overline{b}\times \overline{c}|$

$\therefore \frac{areaof\Delta BRC}{areaof\Delta ABC}=\frac{4}{7}$