Mathematics

Mathematics

Problem

Consider the points P(1,4,−1)P(1, 4, -1), Q(6,−8,7)Q(6, -8, 7), and R(−5,−2,−9)R(-5, -2, -9). Let M be the midpoint of [PQ][PQ], and N be the midpoint of [QR][QR]. a. <b>a</b> Find the coordinates of M and N. b. <b>b</b> Show that [MN][MN] is half the length of [PR][PR] .
Intuitive explanation

The key is recognizing that midpoints are found by averaging coordinates pairwise. Once you find MM and NN, you'll have two specific points in 3D space, which opens the door to finding distances, equations of lines, or other geometric relationships between them. Think of midpoints as "balance points"—they sit exactly halfway along a segment, so each coordinate is simply the average of the endpoints' coordinates.

Intuitive explanation

The key is recognizing that midpoints are found by averaging coordinates pairwise. Once you find MM and NN, you'll have two specific points in 3D space, which opens the door to finding distances, equations of lines, or other geometric relationships between them. Think of midpoints as "balance points"—they sit exactly halfway along a segment, so each coordinate is simply the average of the endpoints' coordinates.

Browse the problem library