Question

calculate the area of triangle abc with altitude cd, given a (-7, -1), b (-1, 5), c (0, 0), and d (-3, 3). (4 points) 9 square units 18 square units 18.5 square units 21 square units

Explanation:

Step1: Find length of base AB

Use distance formula:
$$AB = \sqrt{(x_2-x_1)^2 + (y_2-y_1)^2}$$
$$AB = \sqrt{(-1-(-7))^2 + (5-(-1))^2} = \sqrt{6^2 + 6^2} = \sqrt{72} = 6\sqrt{2}$$

Step2: Find length of altitude CD

Use distance formula:
$$CD = \sqrt{(x_2-x_1)^2 + (y_2-y_1)^2}$$
$$CD = \sqrt{(-3-0)^2 + (3-0)^2} = \sqrt{(-3)^2 + 3^2} = \sqrt{18} = 3\sqrt{2}$$

Step3: Calculate triangle area

Use area formula:
$$Area = \frac{1}{2} \times base \times height$$
$$Area = \frac{1}{2} \times 6\sqrt{2} \times 3\sqrt{2} = \frac{1}{2} \times 36 = 18$$

Answer:

18 square units