The coordinates of the vertices B and C of a triangle ABC are (1, –2) and (2, 3), respectively. Third vertex A lies on the line 2x + y – 2 = 0. The area of the triangle is 8 square units. Find the coordinates of vertex A.

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Points to Remember:

  • The area of a triangle with coordinates (x1, y1), (x2, y2), (x3, y3) is given by: 0.5 * |x1(y2 – y3) + x2(y3 – y1) + x3(y1 – y2)|
  • The equation of a line is given. We need to find a point on this line that satisfies the area constraint.

Introduction:

This question is a problem in coordinate geometry. We are given the coordinates of two vertices of a triangle and the equation of a line on which the third vertex lies. The area of the triangle is also known. The task is to find the coordinates of the third vertex using the given information. This requires applying the formula for the area of a triangle given its vertices and solving a resulting equation.

Body:

1. Setting up the Area Equation:

Let A = (x, y). The coordinates of B and C are (1, -2) and (2, 3) respectively. The area of the triangle ABC is given by:

Area = 0.5 * |x(-2 – 3) + 1(3 – y) + 2(y – (-2))| = 8

Simplifying, we get:

8 = 0.5 * |-5x + 3 – y + 2y + 4|
16 = |-5x + y + 7|

This gives us two possible equations:

Equation 1: -5x + y + 7 = 16 => -5x + y = 9
Equation 2: -5x + y + 7 = -16 => -5x + y = -23

2. Solving for the Coordinates of A:

Since A lies on the line 2x + y – 2 = 0, we have y = 2 – 2x. Substituting this into the two equations above:

For Equation 1:

-5x + (2 – 2x) = 9
-7x = 7
x = -1
y = 2 – 2(-1) = 4
Therefore, A = (-1, 4)

For Equation 2:

-5x + (2 – 2x) = -23
-7x = -25
x = 25/7
y = 2 – 2(25/7) = -36/7
Therefore, A = (25/7, -36/7)

3. Verification:

Let’s verify the area using the coordinates we found:

For A = (-1, 4):

Area = 0.5 * |-1(-2 – 3) + 1(3 – 4) + 2(4 – (-2))| = 0.5 * |5 – 1 + 12| = 0.5 * 16 = 8 (Correct)

For A = (25/7, -36/7):

Area = 0.5 * |(25/7)(-2 – 3) + 1(3 – (-36/7)) + 2(-36/7 – (-2))| = 0.5 * |(-125/7) + (57/7) + (-28/7)| = 0.5 * |-96/7| = 48/7 ≠ 8 (Incorrect)

Conclusion:

By applying the formula for the area of a triangle and the equation of the line on which vertex A lies, we have found two potential coordinates for A. After verification, only A = (-1, 4) satisfies the given area of 8 square units. Therefore, the coordinates of vertex A are (-1, 4). This problem highlights the importance of verifying solutions in coordinate geometry problems, as multiple solutions might arise during the calculation process, but only one will satisfy all given conditions. This approach emphasizes the precise application of geometric principles and algebraic manipulation for solving such problems.

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