When lines y=square root of 3x and y=-square root of 3x +6 intersect at point A, and line y=- square root of 3x +6 and the x axis intersect at point B, solve the following problems. Find the distances OA, OB, and AB
To find point A, we need to set the two equations equal to each other and solve for x:
sqrt(3x) = -sqrt(3x) + 6
2sqrt(3x) = 6
sqrt(3x) = 3
3x = 9
x = 3
Now that we have x=3, we can find the y-coordinate by plugging it back into one of the equations. Let's use y = sqrt(3x):
y = sqrt(3(3))
y = sqrt(9)
y = 3
So point A is at (3, 3).
Now let's find point B where the line y = -sqrt(3x) + 6 intersects the x-axis:
0 = -sqrt(3x) + 6
sqrt(3x) = 6
3x = 36
x = 12
Point B is at (12, 0).
Now we can find the distances:
OA = sqrt((3-0)^2 + (3-0)^2) = sqrt(9 + 9) = sqrt(18) = 3sqrt(2)
OB = sqrt((12-0)^2 + (0-0)^2) = sqrt(144 + 0) = 12
AB = sqrt((12-3)^2 + (0-3)^2) = sqrt(81 + 9) = sqrt(90) = 3sqrt(10)
So,
OA = 3sqrt(2)
OB = 12
AB = 3sqrt(10)