Section Formula

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When a point P divides the line segment joining points A(x₁, y₁) and B(x₂, y₂) internally in the ratio m:n, the coordinates of P are: x = (mx₂ + nx₁)/(m + n), y = (my₂ + ny₁)/(m + n)…

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Internal division occurs when a point P lies on the line segment AB between points A and B, dividing it into two parts AP and PB in a given ratio m:n. The point P is said to divide AB internally in th…

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The coordinates of the midpoint M of a line segment joining points A(x₁, y₁) and B(x₂, y₂) are: M = ((x₁ + x₂)/2, (y₁ + y₂)/2). This is obtained by putting m = n = 1 in the section formula.

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The coordinates of the centroid G of a triangle with vertices A(x₁, y₁), B(x₂, y₂), and C(x₃, y₃) are: G = ((x₁ + x₂ + x₃)/3, (y₁ + y₂ + y₃)/3)…

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Using section formula with m = 1, n = 2: x = (1×6 + 2×2)/(1+2) = (6+4)/3 = 10/3, y = (1×7 + 2×3)/(1+2) = (7+6)/3 = 13/3. Therefore, P = (10/3, 13/3)…

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It means that if point P divides line segment AB, then the ratio of distances AP:PB = m:n. In other words, AP/PB = m/n, where AP and PB are the lengths of the two segments.

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Using midpoint formula: M = ((-3+5)/2, (4+(-2))/2) = (2/2, 2/2) = (1, 1). Therefore, the midpoint is (1, 1).

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The centroid is the point where all three medians of a triangle intersect. It divides each median in the ratio 2:1, with the longer segment being towards the vertex. It is also the center of mass of t…