Answer and Step-by-step explanation:
To calculate the perimeter of the rhombus ABCD, we need to find the lengths of all four sides and then add them together.
Given that point A is (-8, 6) and point D is (3, 9), we can calculate the lengths of sides AB, BC, CD, and DA using the distance formula:
1. Calculate the length of side AB:
- The distance between points A(-8, 6) and B(x, y) is given by the equation:
sqrt((x - (-8))^2 + (y - 6)^2) = sqrt((x + 8)^2 + (y - 6)^2)
2. Calculate the length of side BC:
- Since point C lies on the line y = 3x, we can substitute the x-coordinate of point B into the equation y = 3x to find the y-coordinate of point C.
- The distance between points B(x, y) and C(x, 3x) is given by the equation:
sqrt((x - x)^2 + (y - 3x)^2) = sqrt(0^2 + (y - 3x)^2)
3. Calculate the length of side CD:
- The distance between points C(x, 3x) and D(3, 9) is given by the equation:
sqrt((x - 3)^2 + (3x - 9)^2)
4. Calculate the length of side DA:
- The distance between points D(3, 9) and A(-8, 6) is given by the equation:
sqrt((3 - (-8))^2 + (9 - 6)^2) = sqrt((3 + 8)^2 + (9 - 6)^2)
Once you have calculated the lengths of all four sides, add them together to find the perimeter of the rhombus ABCD. Leave your answer in simplest surd form.