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inscribe a rectangle of base b and height h in a circle of radius one, and inscribe an isosceles triangle in a region of the circle cut off by one base of the rectangle (with that side as the base of the triangle). for what value of h do the rectangle and triangle have the same area?

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  1. let the height of the triangle be y. Then since the center of the rectangle is also the center of the circle,
    y + h/2 = 1
    and so we have
    bh = by/2
    bh = b(1-h/2)/2
    solve for h, and use (b/2)^2 + (h/2)^2 = 1
    to find b

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