The 3-D Coordinate System — Question 4

Question 4

The points $A=(1,0,2)$, $B=(4,4,1)$, and $C=(-1,3,6)$ form a triangle. Classify it as scalene, isosceles, or equilateral, and determine whether it is acute, right, or obtuse.

Question 4 – Solution

Strategy Compare squared side lengths; this avoids unnecessary square roots. The converse of the Pythagorean theorem classifies the largest angle.

Side lengths Direct calculation gives $$AB^2=3^2+4^2+(-1)^2=26,\quad AC^2=(-2)^2+3^2+4^2=29,\quad BC^2=(-5)^2+(-1)^2+5^2=51.$$ The three lengths differ, so the triangle is scalene.

Angle classification The longest side is $BC$. Since $BC^2=51<26+29=55$, the angle opposite $BC$ is acute. The other angles are necessarily smaller, so the triangle is $\boxed{\text{scalene and acute}}$.

Verification The triangle inequality holds; for example, $\sqrt{26}+\sqrt{29}>\sqrt{51}$.