Cartesian coordinates and the straight line — A/L Combined Mathematics

What must I remember about Cartesian coordinates and the straight line?

• Rectangular Cartesian coordinate system: Two mutually perpendicular directed real lines (x-axis and y-axis) intersecting at the Origin O(0, 0), dividing plane into four quadrants (I, II, III, IV).
• Distance formula: Distance d between points P(x1, y1) and Q(x2, y2) is given by Euclidean metric: d = √[(x2 - x1)² + (y2 - y1)²].
• Section formula (Division of line segment): Coordinates of point R dividing line segment PQ in ratio m : n: Internal division: R = ((m*x2 + n*x1)/(m + n), (m*y2 + n*y1)/(m + n)); External division: R = ((m*x2 - n*x1)/(m - n), (m*y2 - n*y1)/(m - n)).
• Midpoint formula: Special case of section formula with m : n = 1 : 1 gives midpoint M = ((x1 + x2)/2, (y1 + y2)/2).
• Centroid of a triangle: The three medians of triangle ABC with vertices (x1, y1), (x2, y2), (x3, y3) concur at Centroid G = ((x1 + x2 + x3)/3, (y1 + y2 + y3)/3), dividing each median in ratio 2 : 1.

Test yourself (1 minute)

1. Area of triangle from vertex coordinates

   Area Δ = (1/2) |x1(y2 - y3) + x2(y3 - y1) + x3(y1 - y2)|; evaluated as determinant; if area = 0, the three points are collinear.

2. Incenter coordinates formula

   Triangle ABC with side lengths a, b, c opposite to vertices A(x1, y1), B(x2, y2), C(x3, y3) has Incenter I = ((a*x1 + b*x2 + c*x3)/(a + b + c), (a*y1 + b*y2 + c*y3)/(a + b + c)).

3. Orthocenter and Circumcenter relationship (Euler line)

   In any non-equilateral triangle, Orthocenter H, Centroid G, and Circumcenter O are collinear on the Euler Line, with G dividing HO in ratio HG : GO = 2 : 1.

Practise Cartesian coordinates and the straight line questions today

Source: NIE G.C.E. (A/L) Combined Mathematics Resource Book - Pure Mathematics, Unit 17

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Where does Cartesian coordinates and the straight line fit in the syllabus?

Year 1 · Term 3