When lines y=x+8 and y=-3x intersect at point A, and line y=x+8 and the x axis intersect at point B, solve the following problems find the distances OA, OB, and AB
To find the coordinates of point A, we first need to find the x-coordinate at which the two lines intersect:
y = x + 8
y = -3x
Setting the two equations equal to each other:
x + 8 = -3x
4x = -8
x = -2
Substitute x = -2 back into one of the equations to find the y-coordinate at point A:
y = -2 + 8
y = 6
Point A is therefore (-2, 6).
To find the coordinates of point B, we set y = x + 8 equal to y = 0 (on the x-axis):
x + 8 = 0
x = -8
Point B is therefore (-8, 0).
Now we can calculate the distances:
Distance OA = √((-2 - 0)^2 + (6 - 0)^2) = √(4 + 36) = √40 = 2√10
Distance OB = |-8 - 0| = 8
Distance AB = √((-2 + 8)^2 + (6 - 0)^2) = √(36 + 36) = √72 = 6√2
Therefore, OA = 2√10, OB = 8, and AB = 6√2.