The base of an isosceles triangle is parallel to x-axis and its end-points belong to the parabola, y=x(10−x) . the vertex of the triangle belongs to the x-axis. find area of the triangle if the length of its base is 8 units.

So, the triangle is an isosceles, two sides are equal, parallel to the x-axis simply means one side is horizontal, say the base, and a vertex of it touches the x-axis.  Check the picture below, the vertex is much higher btw, is at 5, 25, that's just a sketch, and since the function is y = x(10-x), if you zero it out, you get the x-intercepts of 0 and 10.

so.. hmmm if the base is 8 units long, that means, the first point is say at  "x", and the second point will then be 8 units later or "x + 8".

Now, when at those x-coordinates, since the base is horizontal, then the value of "y"is the same, so is the same exact value at f(x) as it'll be at f(x+8), namely the "rise" is 0.

so, we know what f(x) is, hmm let's check what f(x+8) is then.





since we're on the 1st quadrant, it can't be a negative value, so x = 1, well, what's f(1) then?   well is 9.