A line with slope 3 is 4 units away from the origin. Find the area of the triangle formed by this line and the coordinate axes.
+128474 The line will have the form y = 3x + b
In standard form we have 3x - y + b = 0
Using the formula for the distance from a point to a line we can solve for b thusly :
l 3(0) - 0 + b l
______________ = 4
sqrt ( 3^2 + 1^2)
b
________ = 4
sqrt (10)
b = 4 sqrt (10) = the height of the triangle
So the eqation of the line is
y = 3x + 4sqrt (10)
Letting y = 0, we can find the base of the triangle as
3x = -4sqrt (10)
x = -(4/3)sqrt (10)
The absolute value of this is the base length
So....area of the triangle is
(1/2) (base) (height) =
(1/2) (4/3)sqrt (10) (4 sqrt (10) =
(1/2) ( 4/3) ( 4) ( 10) =
80/3 units^2
