Find the area of a triangle bounded by the y-axis, the line f(x)=10-2x, and the line perpendicular to that passes through the origin.

Question

Find the area of a triangle bounded by the y-axis, the line f(x)=10-2x, and the line perpendicular to that passes through the origin. 

Answer 1

There is no clean and easy way to do this problem.

The perpendicular line through the origin has equation y=x/2.

The vertex of the right angle is (4,2)

With the co-ordinates of the vertices you can use the determinant which is simple in this case since one of the points is the origin: | 0 0 1 |

.5 | 4 2 1 | = 10

| 0 5 0|

or you can get the length of the line through the origin from the formula for the distance from a point to a line as 2√5 and then the base from the Pythagorean theorem as √5. Then (1/2) the base times the altitude is also 10.