Home Class 11 Applied Maths Chapter 15: Straight Lines Exercise 15.1: Coordinate Geometry Review
Chapter 15 • Unit VII

Exercise 15.1: Coordinate Geometry Review

Comprehensive guide and practice questions for CBSE Class 11 Applied Mathematics curriculum (2026-27).

15.1 Coordinate Geometry Review: Distance, Section Formulas, and Centroids

In Applied Mathematics, logistics engineering, and GIS spatial mapping, coordinate geometry bridges visual geometry with algebraic computation. Locating warehouses, optimizing transportation delivery routes, and determining market trading catchments begin with fundamental plane coordinate formulas.

Fundamental Formulae & Theorems

  • Distance Formula: The distance \(d\) between points \(A(x_1, y_1)\) and \(B(x_2, y_2)\): \[ d(A, B) = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2} \]
  • Section Formula (Internal Division): The coordinates of point \(P(x, y)\) dividing the line segment joining \(A(x_1, y_1)\) and \(B(x_2, y_2)\) internally in the ratio \(m : n\): \[ x = \frac{m x_2 + n x_1}{m + n}, \quad y = \frac{m y_2 + n y_1}{m + n} \]
  • Section Formula (External Division): If \(P\) divides \(AB\) externally in ratio \(m : n\) (\(m \neq n\)): \[ x = \frac{m x_2 - n x_1}{m - n}, \quad y = \frac{m y_2 - n y_1}{m - n} \]
  • Midpoint Formula: The midpoint \(M\) of \(AB\) (where \(m : n = 1 : 1\)): \[ M = \left( \frac{x_1 + x_2}{2}, \, \frac{y_1 + y_2}{2} \right) \]
  • Centroid of a Triangle: The point of intersection of the medians of \(\Delta ABC\) with vertices \(A(x_1, y_1), B(x_2, y_2), C(x_3, y_3)\): \[ G = \left( \frac{x_1 + x_2 + x_3}{3}, \, \frac{y_1 + y_2 + y_3}{3} \right) \]
  • Area of a Triangle & Collinearity Criterion: \[ \text{Area}(\Delta ABC) = \frac{1}{2} |x_1(y_2 - y_3) + x_2(y_3 - y_1) + x_3(y_1 - y_2)| \] Three points \(A, B, C\) are collinear if and only if: \[ x_1(y_2 - y_3) + x_2(y_3 - y_1) + x_3(y_1 - y_2) = 0 \]

Comprehensive Applied Example: Central Warehouse Hub Placement

Scenario: A supply chain retailer has three distribution centers located at coordinates \(A(2, 5)\), \(B(-4, 9)\), and \(C(8, -2)\) on a regional grid map (in tens of kilometers).

  1. Find the coordinates of the centroid \(G\) to position a centralized dispatch facility.
  2. Determine the land area bounded by the triangle connecting the three distribution centers.
  3. If a new charging station \(P\) divides the highway route between \(A\) and \(C\) internally in the ratio \(2 : 1\), find the coordinates of \(P\).

Solution:

  1. Centroid Coordinates: \[ x_G = \frac{2 + (-4) + 8}{3} = \frac{6}{3} = 2 \] \[ y_G = \frac{5 + 9 + (-2)}{3} = \frac{12}{3} = 4 \] Centroid \(G = (2, 4)\).
  2. Area of Triangle \(ABC\): \[ \text{Area} = \frac{1}{2} |2(9 - (-2)) + (-4)(-2 - 5) + 8(5 - 9)| \] \[ = \frac{1}{2} |2(11) + (-4)(-7) + 8(-4)| = \frac{1}{2} |22 + 28 - 32| = \frac{1}{2} |18| = 9 \text{ sq. units.} \] (Representing \(9 \times 100 = 900\text{ km}^2\)).
  3. Coordinates of Charging Station \(P\): Here \(A(2, 5)\), \(C(8, -2)\), and \(m : n = 2 : 1\): \[ x_P = \frac{2(8) + 1(2)}{2 + 1} = \frac{16 + 2}{3} = \frac{18}{3} = 6 \] \[ y_P = \frac{2(-2) + 1(5)}{2 + 1} = \frac{-4 + 5}{3} = \frac{1}{3} \] Charging Station \(P = \left(6, \frac{1}{3}\right)\).
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