SOLUTION: Show that (1,2), (4,6) and (10,14) are collinear By using the distance formula. And slope!

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Question 563408: Show that (1,2), (4,6) and (10,14) are collinear By using the distance formula.
And slope!
Answer by mananth(16946) About Me  (Show Source):
You can put this solution on YOUR website!
Distance between two points (1,2)(4,6)
x1 y1 x2 y2
d= sqrt%28%28y2-y1%29%5E2%2B%28x2-x1%29%5E2%29
1 2 4 6
d= sqrt%28%286-2%09%29%5E2%2B%284-1%29%5E2%09%29
d= sqrt%28%284%29%5E2%2B%28%093%29%5E2%09%29
d= sqrt%28%2825%29%29
d=5
Distance between two points (10,14)(4,6)

x1 y1 x2 y2
d= sqrt%28%28y2-y1%29%5E2%2B%28x2-x1%29%5E2%29
10 14 4 6
d= sqrt%28%286-14%29%5E2%2B%284-10%29%5E2%29
d= sqrt%28%28-8%29%5E2%09%2B%28-6%29%5E2%09%29
d= sqrt%28%28100%29%29
d= 10.00

Distance between two points (10,14) (1,2)
x1 y1 x2 y2
d= sqrt%28%28y2-y1%29%5E2%2B%28x2-x1%29%5E2%29
10 14 1 2
d= sqrt%28%282-14%29%5E2%2B%281-10%29%5E2%29
d= sqrt%28%28-12%29%5E2%09%2B%28-9%29%5E2%09%29
d= sqrt%28%28225%29%29
d= 15.00
5+10 = 15
A=(4,6), B=(10,14) C=(1,2)
d(BC) = 15
d(AB)=10
d(AC)=5
d(AB) +d(AC)= d(BC)
10+5=15
This is possible only if the points are colinear, else the sum of any two distances will not be equal to the third distance.

so the points are colinear.
BY slopes (1,2)(4,6)
x1 y1 x2 y2
1 2 4 6

slope m =(y2-y1)/(x2-x1)
(6-2)/( 4-1)
( 4 / 3 )
m= 4/ 3

(4,6)(10,14)
x1 y1 x2 y2
4 6 10 14

slope m = (y2-y1)/(x2-x1)
(14-6)/(10-4)
(8/6)
m= 4/ 3
the slopes are same