To find the area of triangle ABO, we first need to find the coordinates of points A and B.
1. To find point A, we need to solve the system of equations y = √(3x) and y = -√(3x) + 6:
√(3x) = -√(3x) + 6
2√(3x) = 6
√(3x) = 3
3x = 9
x = 3
Substitute x = 3 back into y = √(3x) or y = -√(3x) + 6 to find the y-coordinate:
y = √(9) = 3 or y = -√(9) + 6 = -3 + 6 = 3
So, point A is (3, 3).
2. To find point B, we need to solve the system of equations y = -√(3x) + 6 and y = 0:
-√(3x) + 6 = 0
-√(3x) = -6
√(3x) = 6
3x = 36
x = 12
Substitute x = 12 back into y = -√(3x) + 6 to find the y-coordinate:
y = -√(36) + 6 = -6 + 6 = 0
So, point B is (12, 0).
Now, we can find the distance between points A and B using the distance formula:
AB = sqrt[(12-3)^2 + (0-3)^2]
AB = sqrt[9^2 + 3^2]
AB = sqrt[81 + 9]
AB = sqrt(90)
AB = 3√10
The area of triangle ABO can be calculated using the formula for the area of a triangle:
S = (1/2) * base * height
In this case, the base AB = 3√10 and the height is the y-coordinate of point A, which is 3:
S = (1/2) * 3√10 * 3
S = 4.5√10
Therefore, the area of triangle ABO is 4.5√10 square units.
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 area S of triangle ABO
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