Arkimedes for a equation y=x(2-x) and x-axis

[rama1001]

1. Examine what the relationship is by calculating the parabolic segment and the
inscribed triangle bounded by the function
y =-x2 + 2x and the x-axis.

2. Description:
A horizontal line intersects a second degree curve in the points A and B. The line AB and
quadratic curve enclosing an area. This area is called a parabolic segment. it
tangent to the curve which is parallel to the chord AB tangent to the curve in C.
The Greek mathematician, physicist and inventor Archimedes (287-212 BC) discovered
that the area of triangle ABC and the area of ​​the parabolic segment always has the same
relationship.

The Attempt at a Solution

:
I found two pints on x-axis that where the parabola is intersected but the problem comes with finding the third point of the triangle, where a tangent intersect the parabola parallel to the x-axis. The two points are (0,0) and (-2,0).

 

[LightHero]

The third point is (1,1) which is where the tangent line intersects the parabola. This creates a triangle with the area of triangle ABC being 1/2 and the area of the parabolic segment being 4. The relationship between the two is that the area of the triangle is half of the area of the parabolic segment.