Answer:
The perimeter of triangle ABC with altitude BD can be found by using the Distance Formula. The coordinates of the vertices are A(-5, 2), B(-3, 5), C(5, 2), and D(-3, 2). The perimeter is then calculated as the sum of the distances between each pair of points: P = AB + BC + CD + DA = 8.24 units. The area can be found using Heron's Formula, which gives an area of 6.00 units.
Explanation:
The perimeter of triangle ABC with altitude BD can be found by using the Distance Formula. The coordinates of the vertices are A(-5, 2), B(-3, 5), C(5, 2), and D(-3, 2). Using the Distance Formula, the sides of the triangle can be calculated as follows: AB = √((-3 - (-5))² + (5 - 2)²) = √(8 + 9) = √17; BC = √((5 - (-3))² + (2 - 5)²) = √(8 + 9) = √17; and AC = √((-5 - 5)² + (2 - 2)²) = 0. Therefore, the perimeter of triangle ABC is AB + BC + AC = 17√2 units.
The area of triangle ABC can be found using Heron's Formula, which states that the area of a triangle is equal to the square root of s(s-a)(s-b)(s-c), where s is half of the perimeter and a, b, and c are the lengths of each side. In this case, s is half of 17√2 or 8.5√2 units. Plugging in these values into Heron's Formula yields an area for triangle ABC equal to 8.5√2 * 4 * 3 * 1.5√2 or 36√2 units squared.