Vector Algebra
Triangle Area — Largest Side Squared
nta_pyq_2024_apr
Grade 12
Question:
Let $ABC$ be a triangle of area $15\sqrt{2}$ and the vectors $\overrightarrow{AB}=\hat{i}+2\hat{j}-7\hat{k}$, $\overrightarrow{BC}=a\hat{i}+b\hat{j}+c\hat{k}$ and $\overrightarrow{AC}=6\hat{i}+d\hat{j}-2\hat{k}$, $d>0$. Then the square of the length of the largest side of the triangle $ABC$ is _____
Step-by-Step Solution
Key Concept: Area $=\frac{1}{2}|\overrightarrow{AB}\times\overrightarrow{AC}|=15\sqrt{2}$. $\overrightarrow{AB}\times\overrightarrow{AC}=(7d-4)\hat{i}-40\hat{j}+(d-12)\hat{k}$. $(7d-4)^2+1600+(d-12)^2=1800\Rightarrow50d^2-80d-40=0\Rightarrow5d^2-8d-4=0$.
$d=2$. $|AB|^2=54$ is the largest.
Correct Answer: 54