Question

Help please number 9, 10

Answer 1

Answer:

#9, G = (-8,-10)

#10, distance = √122 = 11.0453610171

midpoint = (7/2,7/2)

Step-by-step 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 ✔