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I'd like to ask if someone could help me out with this problem.

Let's have a triangle with coordinates $[0,0],[4,0],[1,3]$. Inscribe a rectangle into this triangle, so its area is maximized The base of rectangle lays on axis $x$.

I know how to proceed if it's a right triangle, but don't know how to proceed now.

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Check this and this. –  metacompactness Jan 13 at 10:27
    
If it is not a right-angled triangle, you will find an optimal rectange has one side on an edge of the triangle, centred at the midpoint of that edge. Solve the question for one edge, and the solution on the other edges will be obvious. –  Henry Jan 13 at 10:40

1 Answer 1

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I assume that you want that the corners of the rectangle be along the sides of the triangles.

The equations of the lines which define the triangle are y = 3 x and y = 4 - x.

Now, let use define four points (x1,0), (x1,y1), (x2,0) and (x2,y2) which will define the rectangle. Since it is a rectangle, y2 = y1. Now, use the equations y1 = 3 x1 and y2 = 4 - x2; since y1 = y2, then 3 x1 = 4 - x2 which can reduce to x2 = 4 - 3 x1.

The area of the rectangle is defined by Area = (x2 - x1) y2 that is to say
Area = (4 - 4 x1) (3 x1) = 12 (1 - x1) x1.

You want this area to be maximized. Compute the derivative of the area with respect to x1 and set it to zero. This will give you the value of x1; from it, x2 = 4 - 3 x1, y = 3 x1 ...

I am sure you can take from here.

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