Question

Statement-$1$: If vertices of a triangle are $A\left(\overrightarrow{a}\right)$ , $B\left(\overrightarrow{b}\right)$ , $C\left(\overrightarrow{c}\right)$ then length of altitude through $A$ is $\frac{|\overrightarrow{a}\times \overrightarrow{b}+\overrightarrow{b}\times \overrightarrow{c}+\overrightarrow{c}\times \overrightarrow{a}|}{|\overrightarrow{b}-\overrightarrow{c}|}$

Statement-$2$: Area of triangle is $\Delta =\frac{1}{2}$ (Base) (Height) $=\frac{1}{2}|\overrightarrow{AB}\times \overrightarrow{AC}|$

• A. Statement$1$ is true, statement-$2$ is true, statement-$2$ is a correct explanation for statement-$1$.
• B. Statement-$1$ is true, statement-$2$ is true, statement-$2$ is not a correct explanation for statement-$1$
• C. Statement-$1$ is true, statement-$2$ is false
• D. Statement-$1$ is false, statement-$2$ is true

Answer 1

The correct option is A Statement$1$ is true, statement-$2$ is true, statement-$2$ is a correct explanation for statement-$1$.

Let $\overrightarrow{AB}\perp \overrightarrow{BC}$ in $△ABC$
Area of $△=\frac{1}{2}|\overrightarrow{AB}||\overrightarrow{BC}|$
$=\frac{1}{2}|\overline{a}\times \overline{b}+\overline{b}\times \overline{c}+\overline{c}\times \overline{a}|=\frac{1}{2}|\overrightarrow{AB}||\overrightarrow{BC}|$
$=\frac{|\overline{a}\times \overline{b}+\overline{b}\times \overline{c}+\overline{c}\times \overline{a}|}{|\overrightarrow{BC}|}=|\overrightarrow{AB}|=$ Length of attitude