Midpoints and ratios
Objective. Divide line segments in given ratios using coordinates.
- 1. Learn
- 2. Worked example
- 3. Practice
- 4. Feedback
- 5. Continue
Learn: the key idea
The midpoint of a line segment joining (x1,y1) and (x2,y2) is ((x1+x2)/2, (y1+y2)/2), the average of the coordinates. To divide a segment in a ratio m:n from point A to point B, find the point A + (m/(m+n))×(vector AB), applying this fraction to both the x and y changes separately. For A(1,2) and B(9,10) divided in ratio 1:3, the point lies 1/4 of the way from A to B: x = 1+(1/4)(9−1)=3, y=2+(1/4)(10−2)=4, giving (3,4). Always identify which point the ratio starts from, since reversing the order changes the answer. Checking the total distance ratio confirms the point divides the segment correctly.
Worked example
Point P divides the line segment from A(2, 3) to B(10, 19) in the ratio 1:3. Find the coordinates of P.
- 1Fraction of the way from A to B: 1/(1+3) = 1/4
- 2x-coordinate: 2 + 1/4 × (10−2) = 2 + 2 = 4
- 3y-coordinate: 3 + 1/4 × (19−3) = 3 + 4 = 7
- 4Point P is (4, 7)
Answer: (4, 7)
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Practice
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Start practisingCommon mistake. Pupils divide a segment in ratio 1:3 by moving 3/4 of the way from A instead of 1/4, confusing which part of the ratio corresponds to the first section. For A to B in ratio 1:3, the correct fraction from A is 1/(1+3)=1/4, not 3/4. Always add the ratio parts to find the correct fraction.
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