Question

If $A(1,2,3)$ , $B(2,5,-1)$ and $C(-1,1,2)$ are the vertices of a triangle, then the area of the triangle is

• A. $\sqrt{155}$
• B. $\frac{1}{2}\sqrt{155}$
• C. $\frac{1}{3}\sqrt{155}$
• D. $\frac{1}{4}\sqrt{155}$

Answer 1

Answer: (B)

$\frac{1}{2}\sqrt{155}$

$A=\hat{i}+2\hat{j}+3\hat{k}$

$B=\hat{i}+5\hat{j}-\hat{k}$

$C=-\hat{i}+\hat{j}+2\hat{k}$

Therefore

$\overrightarrow{AB}=\hat{i}+3\hat{j}-4\hat{k}$

$\overrightarrow{BC}=-3\hat{i}-4\hat{j}+3\hat{k}$

Therefore area is

$=\frac{1}{2}|\overrightarrow{AB}\times \overrightarrow{BC}|$

$=\frac{1}{2}|-7\hat{i}+9\hat{j}+5\hat{k}|$

$=\frac{1}{2}\sqrt{49+81+25}$

$=\frac{155}{2}$ sq units.