Triangle G B H and triangle A B C with points A (3, negative 9); B (3, 1); and C (13, 1). Point G lies on segment A B, and point H lies on segment C B. Which information about the coordinates of G and H will prove that GH¯¯¯¯¯¯¯¯ is a midsegment of △ABC?
A. G(1.5,5); H(8,5)
B. G(−4,3); H(1,8)
C. G(6,−8); H(16,8)
D. G(3,−4); H(8,1)
The midsegment of a triangle connects the midpoint of two sides of the triangle. In this case, GH¯¯¯¯¯¯¯¯ is the midsegment connecting the midpoint of side AB (point G) and the midpoint of side BC (point H).
The midpoint of side AB can be found by averaging the x-coordinates and y-coordinates of points A and B:
Midpoint of AB:
(x-coordinate) = (3 + 3) / 2 = 6 / 2 = 3
(y-coordinate) = (-9 + 1) / 2 = -8 / 2 = -4
Therefore, the midpoint of AB is (3, -4).
Similarly, the midpoint of side BC can be found by averaging the x-coordinates and y-coordinates of points B and C:
Midpoint of BC:
(x-coordinate) = (3 + 13) / 2 = 16 / 2 = 8
(y-coordinate) = (1 + 1) / 2 = 2 / 2 = 1
Therefore, the midpoint of BC is (8, 1).
To prove that GH¯¯¯¯¯¯¯¯ is a midsegment, the coordinates of G and H should match the calculated midpoints.
Checking the options:
A. G(1.5,5); H(8,5) - Incorrect coordinates for G, correct coordinates for H
B. G(−4,3); H(1,8) - Incorrect coordinates for G and H
C. G(6,−8); H(16,8) - Incorrect coordinates for G and H
D. G(3,−4); H(8,1) - Correct coordinates for G and H
Only option D, G(3,−4); H(8,1), matches the calculated midpoints. Therefore, option D is the correct answer.