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 $K M$

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Answer

$b)$ PQ is half the length of KM because the length of PQ is 5 and the length of KM is 10

Steps

Step 1 ：Find the coordinates of points P and Q: $P = (\frac{2 + 4}{2}, \frac{6 + 10}{2}) = (3, 8)$ and $Q = (\frac{4 + 8}{2}, \frac{10 - 2}{2}) = (6, 4)$

Step 2 ：Find the slopes of PQ and KM: $s l o p e_{P Q} = \frac{4 - 8}{6 - 3} = - \frac{4}{3}$ and $s l o p e_{K M} = \frac{- 2 - 6}{8 - 2} = - \frac{8}{6} = - \frac{4}{3}$

Step 3 ：Find the lengths of PQ and KM: $l e n g t h_{P Q} = \sqrt{(6 - 3)^{2} + (4 - 8)^{2}} = \sqrt{9 + 16} = 5$ and $l e n g t h_{K M} = \sqrt{(8 - 2)^{2} + (- 2 - 6)^{2}} = \sqrt{36 + 64} = 10$

Step 4 ： $a)$ PQ is parallel to KM because their slopes are equal $- \frac{4}{3}$

Step 5 ： $b)$ PQ is half the length of KM because the length of PQ is 5 and the length of KM is 10

2) The vertices of a triangle are K(2,6),L(4,10), and M(8,−2). Let P be the midpoint of KL and Q be the midpoint of LM. Verify that...a) PQ is parallel to KMb) PQ is half the length of KM

Answer

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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 $K M$