Question

Find the area of the triangle whose vertices are (-5, -1), (3, -5) and (5, 2).

Hint:

To avoid mistakes with signs, calculate each part of the formula \(x_1(y_2 - y_3)\), \(x_2(y_3 - y_1)\), and \(x_3(y_1 - y_2)\) separately before adding them together. The absolute value at the end ensures that the area is always a positive quantity.

Answer 1

Step 1: Understanding the Concept:
To find the area of a triangle when the coordinates of its three vertices are given, we use the coordinate geometry formula for the area of a triangle.

Step 2: Key Formula or Approach:
The area of a triangle with vertices \((x_1, y_1)\), \((x_2, y_2)\), and \((x_3, y_3)\) is given by the formula: \[ \text{Area} = \frac{1}{2} |x_1(y_2 - y_3) + x_2(y_3 - y_1) + x_3(y_1 - y_2)| \]

Step 3: Detailed Explanation:
Let the given vertices be: \( (x_1, y_1) = (-5, -1) \)
\( (x_2, y_2) = (3, -5) \)
\( (x_3, y_3) = (5, 2) \)
Substitute these values into the area formula: \[ \text{Area} = \frac{1}{2} |(-5)(-5 - 2) + 3(2 - (-1)) + 5(-1 - (-5))| \] Simplify the terms inside the absolute value: \[ \text{Area} = \frac{1}{2} |(-5)(-7) + 3(2 + 1) + 5(-1 + 5)| \] \[ \text{Area} = \frac{1}{2} |35 + 3(3) + 5(4)| \] \[ \text{Area} = \frac{1}{2} |35 + 9 + 20| \] \[ \text{Area} = \frac{1}{2} |64| \] \[ \text{Area} = \frac{1}{2} \times 64 \] \[ \text{Area} = 32 \]

Step 4: Final Answer:
The area of the triangle is 32 square units.