Please give an explanation to go with your answer! A rectangle is placed symmetrically inside a square. The rectangle has sides of length m and n. Find the area of the square in terms of m and n.​

Accepted Solution

Answer:[tex]\large\boxed{A=\dfrac{(m+n)^2}{2}}[/tex]Step-by-step explanation:Look at the picture.We have the triangles 45° - 45° - 90°. The sides are in ratio 1 : 1 : √2(look at the second picture).Therefore we have the equations:[tex]x\sqrt2=m[/tex]  and  [tex]y\sqrt2=n[/tex]Solve:[tex]x\sqrt2=m[/tex]                multiply both sides by √2[tex]2x=m\sqrt2[/tex]               divide both sides by 2[tex]x=\dfrac{m\sqrt2}{2}[/tex][tex]y\sqrt2=n[/tex]                multiply both sides by √2[tex]2y=n\sqrt2[/tex]             divide both sides by 2[tex]y=\dfrac{n\sqrt2}{2}[/tex]The side length of square is[tex]x+y=\dfrac{m\sqrt2}{2}+\dfrac{n\sqrt2}{2}=\dfrac{m\sqrt2+n\sqrt2}{2}=\dfrac{\sqrt2}{2}(m+n)[/tex]The area of a square:[tex]A=\left(\dfrac{\sqrt2}{2}(m+n)\right)^2=\left(\dfrac{\sqrt2}{2}\right)^2(m+n)^2=\dfrac{2}{4}(m+n)^2=\dfrac{(m+n)^2}{2}[/tex]