Answer:
#9, G = (-8,-10)
#10, distance = √122 = 11.0453610171
midpoint = (7/2,7/2)
Explanation:
#9: We don't need no stinkin' diagrams!!
midpoint M=(-6,-3)
endpoint H=(-4,4)
displacement vector D from H to M:
D = (M-H) = (-6 - -4, -3 - 4) = (-2,-7)
Check: start at endpoint H, move along displacement vector D, should bring you to M.
H + D = (-4,4)+(-2,-7)
= (-4-2,4-7) = (-6,-3) = M ✔
Adding D to endpoint H gets to midpoint M. Adding 2D to H will reach the other endpoint, G.
G = H + 2D = (-4,4)+2×(-2,-7)
= (-4-2×2,4-2×7) = (-8,-10)
Check: Subtracting D from G should bring you back to M.
G-D = (-8,-10) - (-2,-7)
= (-8 - -2, -10 - -7)
= (-6,-3) = M ✔
#10 distance between P and Q, and midpoint of segment PQ:
P=(3,-2) Q=(4,9)
displacement vector D from P to Q:
D = Q - P = (4 - 3, 9 - -2) = (1,11)
so that adding D to P gives Q:
D + P = (1+3,11-2) = (4,9) ✔
Distance between two points is square root of dot product of displacement vector with itself:
d = √(D dot D) = √((1,11)dot(1,11))
= √(1×1+11×11) = √122 = 11.0453610171
Midpoint M is P + D/2, start at P and move half way to Q,
M = (3,-2)+(1,11)/2
= (3+1/2,-2+11/2)
= (7/2,7/2)
Start at M, move D/2 brings you to Q,
M + D/2 = (7/2+7/2) + (1/2,11/2)
= (8/2,18/2) = (4,9) = Q ✔