Inscribed rectangle of given perimeter

Text

Consider the circle of radius $3$ and the inscribed rectangle. Determine the sides of the rectangle if you want it to have perimeter $15$.

Solution

The diagonal is the diameter: $a^2+b^2=36$. From the perimeter $2(a+b)=15$ we get $a+b=\dfrac{15}{2}$. Then $2ab=(a+b)^2-(a^2+b^2)=\frac{225}{4}-36=\frac{81}{4}\ \Rightarrow\ ab=\frac{81}{8},$ $(a-b)^2=36-\frac{81}{4}=\frac{63}{4}\ \Rightarrow\ a-b=\frac{3\sqrt7}{2}.$ Hence $a=\frac{15+3\sqrt7}{4}\approx5{,}73,\qquad b=\frac{15-3\sqrt7}{4}\approx1{,}77.$ $\boxed{a=\frac{15+3\sqrt7}{4}\approx5{,}73,\qquad b=\frac{15-3\sqrt7}{4}\approx1{,}77.}$