Final answer:
To determine the area and perimeter of triangle ABC, calculate the length of each side using the distance formula. Sum these lengths to get the perimeter and apply the area formula, 1/2 × base × height, using the base AC and the vertical distance between A and B or B and C as the height.
Step-by-step explanation:
To find the area and perimeter of triangle ABC with vertices A(-75, -8), B(4, 4), and C(5, -8), we need to use the distance formula to calculate the lengths of the sides of the triangle and then use these lengths to calculate the perimeter and the area.
Calculating the Sides of Triangle ABC
First, use the distance formula d = √((x2 - x1)² + (y2 - y1)²) to find the lengths of AB, BC, and AC:
- AB = √((4 - (-75))² + (4 - (-8))²) = √(6241 + 144) = √(6385)
- BC = √((5 - 4)² + (-8 - 4)²) = √(1 + 144) = √(145)
- AC = √((5 - (-75))² + (-8 - (-8))²) = √(6400 + 0) = √(6400)
Next, calculate the perimeter by adding the lengths of the sides:
Perimeter = AB + BC + AC
Calculating the Area of Triangle ABC
For the area, you can use the formula Area = 1/2 × base × height. In this case, using points A and C for the base and the difference in the y-coordinates of A and B or B and C for the height:
Area = 1/2 × AC × (yB - yA) = 1/2 × 80 × 12 = 480 square units