Calculate Triangle Area from Coordinates
We need to find the area of the triangle with vertices \(A(8, 8)\), \(B(22, 8)\), and \(C(16, -10)\).
Step 1: Identify the Base
Observe the coordinates of vertices A and B. Both have the same y-coordinate (\(y=8\)). This means the side AB is a horizontal line.
- Vertex A: \((x_1, y_1) = (8, 8)\)
- Vertex B: \((x_2, y_2) = (22, 8)\)
- Vertex C: \((x_3, y_3) = (16, -10)\)
The length of the base AB can be calculated as the difference in the x-coordinates:
Base \(AB = |x_2 - x_1| = |22 - 8| = |14| = 14\) units.
Step 2: Calculate the Height
The height of the triangle corresponding to the base AB is the perpendicular distance from vertex C to the horizontal line \(y=8\) (which contains AB).
Height \(h = |y_3 - y_{AB}| = |-10 - 8| = |-18| = 18\) units.
Step 3: Compute the Area
Use the formula for the area of a triangle: Area \(= \frac{1}{2} \times \text{base} \times \text{height}\).
Area \(= \frac{1}{2} \times 14 \times 18\)
Area \(= 7 \times 18\)
Area \(= 126\) square units.
Result
The area of the triangle is 126 square units.
