Coordinate Geometry
About 6 min · Mathematical Reasoning · English
Time worked out from the lesson's 367 words at 150 words a minute, plus a minute for each question.
Coordinate Geometry forms the bridge between algebra and geometry by representing geometric figures using algebraic equations on the Cartesian plane. In SOF IMO, this topic tests your ability to visualize points, calculate distances, and find positions that divide line segments in given ratios.
Key ideas
- Cartesian Plane: A two-dimensional plane defined by perpendicular x-axis (horizontal) and y-axis (vertical) intersecting at origin O(0,0). Any point is represented as an ordered pair (x, y) where x is the abscissa and y is the ordinate.
- Quadrants: The plane divides into four quadrants — I (+,+), II (−,+), III (−,−), IV (+,−). The signs of coordinates determine which quadrant a point lies in. Points on axes belong to no quadrant.
- Distance Formula: The distance between two points A(x₁, y₁) and B(x₂, y₂) is given by d = √[(x₂−x₁)² + (y₁−y₂)²]. This comes from the Pythagorean theorem applied to the right triangle formed by the points.
- Section Formula (Internal Division): If point P divides the line segment joining A(x₁, y₁) and B(x₂, y₂) internally in ratio m:n, then P = ((mx₂+nx₁)/(m+n), (my₂+ny₁)/(m+n)).
- Midpoint Formula: Special case of section formula when m:n = 1:1. Midpoint M = ((x₁+x₂)/2, (y₁+y₂)/2). This is the arithmetic mean of the coordinates.
Quick reference
- Distance: d = √[(x₂−x₁)² + (y₂−y₁)²] — always take square root at the end
- Midpoint: Average both coordinates — ((x₁+x₂)/2, (y₁+y₂)/2)
- Internal division m:n: Weighted average — numerator has cross-multiplication (mx₂+nx₁), denominator is (m+n)
- External division m:n: Use subtraction — (mx₂−nx₁)/(m−n)
- Quadrant signs: I(+,+), II(−,+), III(−,−), IV(+,−)
Check yourself: 3 questions
1. Point A lies on the x-axis at a distance of 5 units from the origin, and point B lies on the y-axis at a distance of 12 units from the origin. Both A and B are in the positive direction. What is the distance between points A and B?
2. The point P divides the line segment joining points A(2, 3) and B(8, 9) in the ratio 1:2. What are the coordinates of point P?
3. Three vertices of a parallelogram ABCD are A(1, 2), B(4, 3) and C(6, 6). What are the coordinates of the fourth vertex D?
This lesson is cut from the topic's note, not rewritten. Fact-checked on 21 Oct 2026: an AI-assisted check of this note against cited sources; the errors it found were corrected that day. Some errors may remain. The questions are from Shishya's validated question bank.
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