Question

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.​

Answer 1

Answer:

$\large\boxed{A=\dfrac{(m+n)^2}{2}}$

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:

$x\sqrt2=m$  and  $y\sqrt2=n$

Solve:

$x\sqrt2=m$                multiply both sides by √2

$2x=m\sqrt2$               divide both sides by 2

$x=\dfrac{m\sqrt2}{2}$

$y\sqrt2=n$                multiply both sides by √2

$2y=n\sqrt2$             divide both sides by 2

$y=\dfrac{n\sqrt2}{2}$

The side length of square is

$x+y=\dfrac{m\sqrt2}{2}+\dfrac{n\sqrt2}{2}=\dfrac{m\sqrt2+n\sqrt2}{2}=\dfrac{\sqrt2}{2}(m+n)$

The area of a square:

$A=\left(\dfrac{\sqrt2}{2}(m+n)\right)^2=\left(\dfrac{\sqrt2}{2}\right)^2(m+n)^2=\dfrac{2}{4}(m+n)^2=\dfrac{(m+n)^2}{2}$