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Homework Statement
Triangle OAB is an isosceles triangle with vertex O at the origin and vertices A and B on the parabola y = 9-x^2
Express the area of the triangle as a function of the x-coordinate of A.
Homework Equations
A = 1/2 bh
Distance formula (maybe)
Heron's Formula (an alternative)
The Attempt at a Solution
The area of a triangle is given by A = 1/2 bh. If point A is (x, 9 - x^2), then b=2x and h = 9-p^2
The final equation will be A = 9x - x^3 (as a function of the x coordinate of A)
The equation above works only for a limited number of triangles embedded in the parabola. For instance, if A and B has the same x coordinates, then the equation will work. But i considered two different triangles that can make an isosceles triangle:
O = (0,0)
A = (2,5)
B = (2,5)
Area is 10 unites^2
and
O = (0,0)
A = (2,5)
B = (-2.2777902,3.8111609)
And the area will be around 4.75 units^2
Both of those are isosceles triangles, that can fit into the parabola of 9-x^2. How would I go about finding an equation that can find the area of all possible isosceles triangles in the parabola?
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