Find the area of the triangle with vertices A(1,1,2), B(2,3,5) and C(1,5,5).
Answer & explanation
Correct answer: option 1
The vertices of the triangle ABC is given as A(1,1,2), B(2,3,5) and C(1,5,5)
The adjacent sides \(\vec{AB}\) and \(\vec{BC}\) of triangle ABC are given as:
\(\vec{AB}\) = (2-1)\(\hat{i}\) + (3-1)\(\hat{j}\) + (5-2) \(\hat{k}\) =\(\hat{i}\) + 2\(\hat{j}\) + 3 \(\hat{k}\)
\(\vec{BC}\)= (1-2)\(\hat{i}\) + (5-3)\(\hat{j}\)+ (5-5)\(\hat{k}\) = -\(\hat{i}\) + 2\(\hat{j}\)
Area of the triangle ABC = 1/2 {|\(\vec{AB}\)x\(\vec{BC}\)|}
\(\vec{AB}\)x\(\vec{BC}\) =( \(\hat{i}\) + 2\(\hat{j}\)+ 3 \(\hat{k}\))x (-\(\hat{i}\)+ 2\(\hat{j}\) + 0\(\hat{k}\))
= (-6) \(\hat{i}\) - 3\(\hat{j}\) + (2+2)\(\hat{k}\)
= -6 \(\hat{i}\) - 3\(\hat{j}\)+ 4\(\hat{k}\)
so, |\(\vec{AB}\)x\(\vec{BC}\)| = √(-6)2+ (-3)2 +(4)2 = √61
Hence, Area of the triangle ABC = √(61)/2