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.

Question

Find the area S of triangle ABO

Answer 1

To find the coordinates of point A, set the two equations equal to each other:
√(3x) = -√(3x) + 6
2√(3x) = 6
√(3x) = 3
3x = 9
x = 3

Substitute x = 3 into either equation to find the y-coordinate of point A:
y = √(3(3)) = 3

Therefore, point A is at (3,3).

To find the coordinates of point B, set y = 0 in the equation y = -√(3x) + 6:
0 = -√(3x) + 6
√(3x) = 6
3x = 36
x = 12

Therefore, point B is at (12,0).

Now, find the distances OA, OB, and AB using the distance formula:
Distance OA = √[(3-0)^2 + (3-0)^2] = √(9+9) = √18 = 3√2
Distance OB = √[(12-0)^2 + (0-0)^2] = √(144) = 12
Distance AB = √[(12-3)^2 + (0-3)^2] = √(81+9) = √90 = 3√10

Lastly, find the area of triangle ABO using the formula for the area of a triangle:
S = (1/2) * base * height
S = (1/2) * OA * OB
S = (1/2) * 3√2 * 12
S = 18√2

Therefore, the area of triangle ABO is 18√2.