2) The vertices of a triangle are $K(2,6), L(4,10)$, and $M(8,-2)$. Let $P$ be the midpoint of $K L$ and $Q$ be the midpoint of LM. Verify that...
a) $P Q$ is parallel to $K M$
b) $P Q$ is half the length of $K M$
Answer
\(\boxed{b)}\) PQ is half the length of KM because the length of PQ is 5 and the length of KM is 10
Step 1 ：Find the coordinates of points P and Q: \(P = \left(\frac{2+4}{2}, \frac{6+10}{2}\right) = (3, 8)\) and \(Q = \left(\frac{4+8}{2}, \frac{10-2}{2}\right) = (6, 4)\)
Step 2 ：Find the slopes of PQ and KM: \(slope_{PQ} = \frac{4-8}{6-3} = -\frac{4}{3}\) and \(slope_{KM} = \frac{-2-6}{8-2} = -\frac{8}{6} = -\frac{4}{3}\)
Step 3 ：Find the lengths of PQ and KM: \(length_{PQ} = \sqrt{(6-3)^2 + (4-8)^2} = \sqrt{9+16} = 5\) and \(length_{KM} = \sqrt{(8-2)^2 + (-2-6)^2} = \sqrt{36+64} = 10\)
Step 4 ：\(\boxed{a)}\) PQ is parallel to KM because their slopes are equal \(-\frac{4}{3}\)
Step 5 ：\(\boxed{b)}\) PQ is half the length of KM because the length of PQ is 5 and the length of KM is 10
