Question

The endpoints of the side of a square lie on the circle and the opposite side of the square tangent to the circle at its midpoint. Find the ratio of the side and the radius.

Answer 1

Answer:

$\displaystyle \frac{L}{r}=\frac{8}{5}$

Step-by-step explanation:

Circle and Square

We have a geometric construction as shown in the image below. We can see that

$\displaystyle r+h=L$

Or, equivalently

$\displaystyle h=L-r$

The triangle formed by r,h and L/2 is right, because the opposite side of the square is tangent to the circle at its midpoint. This means we can use Pythagoras's theorem:

$\displaystyle r^2=h^2+\left(\frac{L}{2}\right)^2$

Replacing h

$\displaystyle r^2=(L-r)^2+\left(\frac{L}{2}\right)^2$

Expanding squares

$\displaystyle r^2=L^2-2Lr+r^2+\frac{L^2}{4}$

Simplifying

$\displaystyle 2Lr=L^2+\frac{L^2}{4}$

Multiplying by 4

$\displaystyle 8r=4L+L$

Joining terms

$\displaystyle 8r=5L$

Solving for the ratio L/R as required

$\displaystyle \frac{L}{r}=\frac{8}{5}$