Let's begin by computing the current centroid, point O_1O1,
O_(1x) = (6 + 5 - 5)/3 = 2O1x=6+5−53=2
O_(1y) = (2 - 8 + 3)/3 = -1O1y=2−8+33=−1
Point O_1 = (2, -1)O1=(2,−1)
- Scale point (6, 2)(6,2):
Compute the vector from point (7, -2)(7,−2) to point (6, 2)(6,2):
(6 - 7)hati + (2 - -2)hatj = -hati + 4hatj(6−7)ˆi+(2−−2)ˆj=−ˆi+4ˆj
Scale by 5:
-5hati + 20hatj−5ˆi+20ˆj
Find the new end point:
(-5+ 7, 20 + -2) = (2, 18)(−5+7,20+−2)=(2,18)
- Scale point (5, -8)(5,−8):
Compute the vector from point (7, -2)(7,−2) to point (5, -8)(5,−8):
(5 - 7)hati + (-8 - -2)hatj = -2hati -6hatj(5−7)ˆi+(−8−−2)ˆj=−2ˆi−6ˆj
Scale by 5:
-10hati - 30hatj−10ˆi−30ˆj
Find the new end point:
(-10+ 7, -30 + -2) = (-3, -32)(−10+7,−30+−2)=(−3,−32)
- Scale point (-5, 3)(−5,3):
Compute the vector from point (7, -2)(7,−2) to point (-5, 3)(−5,3):
(-5 - 7)hati + (3 - -2)hatj = -12hati + 5hatj(−5−7)ˆi+(3−−2)ˆj=−12ˆi+5ˆj
Scale by 5:
-60hati + 25hatj−60ˆi+25ˆj
Find the new end point:
(-60+ 7, 25 + -2) = (-53, 23)(−60+7,25+−2)=(−53,23)
Our scaled triangle has the vertices, (2, 18)(2,18), (-3, -32)(−3,−32), and (-53, 23)(−53,23). The new centroid, O_2O2, has coordinates:
O_(2x) = (2 - 3 - 53)/3 = -18O2x=2−3−533=−18
O_(2y) = (18 - 32 + 23)/3 = 3O2y=18−32+233=3
O_2 = (-18, 3)O2=(−18,3)
Compute the distance, d, between O_1O1 and O_2O2:
d = sqrt((-18 - 2)^2 + (3 - -1)^2)d=√(−18−2)2+(3−−1)2
d = sqrt((-20)^2 + 4^2)d=√(−20)2+42
d = sqrt((-20)^2 + 4^2)d=√(−20)2+42
d = sqrt(416)d=√416
d = 4sqrt(26)d=4√26
