The area of a triangle ABC, whose coordinates are A(x 1, y 1), B(x 2, y 2), C(x 3, y 3) is given by______.
½ [x 1(y 2- y 3) + x 2(y 3 - y 1) + x 3(y 1- y 2)]
Understanding the Area of a Triangle with Coordinates
The area of a triangle can be calculated using the coordinates of its vertices. If the vertices of a triangle ABC are given as \(A(x_1, y_1)\), \(B(x_2, y_2)\), and \(C(x_3, y_3)\), there is a specific formula derived from coordinate geometry to find its area.
The Formula for Triangle Area Using Coordinates
The standard formula for the area of a triangle with vertices \((x_1, y_1)\), \((x_2, y_2)\), and \((x_3, y_3)\) is:
\[ \text{Area} = \frac{1}{2} |x_1(y_2 - y_3) + x_2(y_3 - y_1) + x_3(y_1 - y_2)| \]The absolute value is used because the area must be a non-negative value. However, the expression inside the absolute value sign is often presented as the formula structure, which can be positive or negative depending on the order in which the vertices are taken (clockwise or counterclockwise). The magnitude of this expression, multiplied by \(\frac{1}{2}\), gives the area.
Comparing the Formula with Options
Let's look at the provided options and compare them with the standard formula structure:
- \[ \frac{1}{2} [x_1(y_2 - y_3) + x_2(y_3 - y_1) + x_3(y_1 - y_2)] \] : This option matches the expression inside the absolute value in the standard formula, multiplied by \(\frac{1}{2}\). This is the correct structure for calculating the area using coordinates.
- \[ 2 [x_1(y_2 - y_3) + x_2(y_3 - y_1) + x_3(y_1 - y_2)] \] : This option uses a multiplier of 2 instead of \(\frac{1}{2}\), which is incorrect for the area formula.
- \[ y_1(x_2 - x_3) + y_2(x_3 - x_1) + y_3(x_1 - x_2) \] : This expression is also related to the coordinates but is not the formula for the area. It is related to the condition for collinearity of three points.
- \[ \frac{1}{2} [x_1 y_1 + x_2 y_2 + x_3 y_3] \] : This option presents an incorrect combination of coordinates for calculating the area of a triangle.
Based on the comparison, the first option correctly represents the formula for the area of a triangle given the coordinates of its vertices.
Summary of the Area Formula
The area of a triangle with vertices \( (x_1, y_1) \), \( (x_2, y_2) \), and \( (x_3, y_3) \) is calculated using the determinant-like structure:
\[ \text{Area} = \frac{1}{2} (x_1(y_2 - y_3) + x_2(y_3 - y_1) + x_3(y_1 - y_2)) \]Remember to take the absolute value of the result to ensure the area is positive.
Important Questions from Co-ordinate Geometry
In which ratio the point (-3, p) divides the line segment joining the points (-5, -4) and (-2, 3)?
The area (in sq. units) of the triangle formed by the graphs of 8x + 3y = 24, 2x + 8 = y and the x-axis is:
In which quadrant both abscissa and ordinate are negative?
Find the slope of the line joining the points (3, -4) and (5, 2).
Find the value of K for which equation x – Ky = 2, 3x + 2y = 5 has unique solution.
