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Area of Triangle from Coordinates

XAT 2023 · Quantitative Ability & Data Interpretation · Medium · Coordinate Geometry

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ABC is a triangle and the coordinates of A, B and C are (a,b−2c)(a, b-2c), (a,b+4c)(a, b+4c) and (−2a,3c)(-2a,3c) respectively where aa, bb and cc are positive numbers.

The area of the triangle ABC is:

  1. A.

    6abc

  2. B.

    9abc

  3. C.

    6bc

  4. D.

    9ac

  5. E.

    None of the above

Answer

D

Explanation

Base ABAB lies on line x=ax = a. Length of base AB=∣(b+4c)−(b−2c)∣=6cAB = |(b+4c) - (b-2c)| = 6c. Height from C(−2a,3c)C(-2a, 3c) to line x=ax = a is ∣a−(−2a)∣=3a|a - (-2a)| = 3a. Area = 12×base×height=12×6c×3a=9ac\frac{1}{2} \times \text{base} \times \text{height} = \frac{1}{2} \times 6c \times 3a = 9ac.

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