Inscribed rectangle of given area

Text

A rectangle $ABCD$ is inscribed in a circle of radius $3$. Determine, justifying it with algebraic reasoning too, the sides of the rectangle when its area equals $14$.

Solution

An inscribed rectangle has the diameter as its diagonal: $d=2\cdot3=6$. With sides $a,b$, by Pythagoras $a^2+b^2=36$; also $ab=14$. Using the identities: $(a+b)^2=36+28=64\ \Rightarrow\ a+b=8,\qquad (a-b)^2=36-28=8\ \Rightarrow\ a-b=2\sqrt2.$ Hence $a=4+\sqrt2\approx5{,}41,\qquad b=4-\sqrt2\approx2{,}59.$ $\boxed{a=4+\sqrt2\approx5{,}41,\qquad b=4-\sqrt2\approx2{,}59.}$