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Math Help - Maximise the lengths of two sides of a triangle within a circle

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    Maximise the lengths of two sides of a triangle within a circle

    I'm doing a bunch of analysis problems and have come across this one.
    I feel quite dumb for this, I'm sure this is a 16-year old teenager level basic calculus problem but can't for the life of me remember how to actually formulate the problem.

    I have a triangle within a circle (which I'm calling C) of radius 1 which can be seen in the following diagram.
    The angle TSP is a right angle.
    Maximise the lengths of two sides of a triangle within a circle-photo.jpg


    What is the maximum of the sum of the lengths TS and SP? Presumably, the points S and T can move about providing S stays on the diametre and T stays on the "rim" of the circle and the angle remains as a right angle.
    Ie: Find max(TS+SP).

    My instinct tells me I need to formulate this into a problem featuring something I can differentiate. Then I just find where the derivative equals zero and go from there, like I said; basic calculus principles. But I'm not even sure how to formulate it into an equation.

    Thanks in advance for any assistance y'all can provide.
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    Re: Maximise the lengths of two sides of a triangle within a circle

    Set up a coordinate system so that Q and P are on the x-axis and the center of the circle is the origin. Then the circle is given by equation x^2+ y^2= R^2 where "R" is the length of a radius of the circle (which you don't give but clearly the distance you want depend on R). Take point S to have coordinates (x,0). Then T has coordinates (x, \sqrt{R^2- x^2}) and P has coordinates (R, 0). The distance from S to T is \sqrt{R^2- x^2} and the distance from S to P is R- x. So your total distance is given by R- x+ \sqrt{R^2- x^2}.
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    Re: Maximise the lengths of two sides of a triangle within a circle

    Quote Originally Posted by HallsofIvy View Post
    Set up a coordinate system so that Q and P are on the x-axis and the center of the circle is the origin. Then the circle is given by equation x^2+ y^2= R^2 where "R" is the length of a radius of the circle (which you don't give but clearly the distance you want depend on R). Take point S to have coordinates (x,0). Then T has coordinates (x, \sqrt{R^2- x^2}) and P has coordinates (R, 0). The distance from S to T is \sqrt{R^2- x^2} and the distance from S to P is R- x. So your total distance is given by R- x+ \sqrt{R^2- x^2}.
    Ah yes, thankyou, that makes a lot of sense.

    The radius is 1.
    Following this I let the sum equal f(x) and differentiated, let the derivative equal to zero and obtained the stationary points x=\pm \frac{1}{\sqrt{2}}
    Then  x=\frac{-1}{\sqrt{2}} yields the maximum sum which is 1+\sqrt{2}. (And of course the other root yields the minimum which is 1.)

    Cheers for helping.
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