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Distance and dividing a line
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Two formulae find the length of a line segment and the point that splits it in any ratio.
Distance
The distance between (x1, y1) and (x2, y2) is
√[(x2 − x1)2 + (y2 − y1)2]
(1, 2) to (7, 10): √(36 + 64) = 10
√[(x2 − x1)2 + (y2 − y1)2]
(1, 2) to (7, 10): √(36 + 64) = 10
Midpoint
((x1 + x2)2, (y1 + y2)2)
Midpoint of (−4, 7) and (6, −1) is (1, 3).
Midpoint of (−4, 7) and (6, −1) is (1, 3).
Dividing in a ratio
The point dividing A(x1, y1) to B(x2, y2) in the ratio m : n is
((nx1 + mx2)(m + n), (ny1 + my2)(m + n))
A(1, 2), B(10, 14), ratio 1 : 2: (2 + 103, 4 + 143) = (4, 6)
((nx1 + mx2)(m + n), (ny1 + my2)(m + n))
A(1, 2), B(10, 14), ratio 1 : 2: (2 + 103, 4 + 143) = (4, 6)
Find the distance between (−3, 4) and (2, −8).
- Differences: 5 and −12
- √(25 + 144) = √169
Answer: 13
Distance = √[(x2 − x1)2 + (y2 − y1)2]. The point dividing AB in m : n is A + m(m + n) of the way to B.
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