Answer:
[tex]\displaystyle \frac{L}{r}=\frac{8}{5}[/tex]
Step-by-step explanation:
Circle and Square
We have a geometric construction as shown in the image below. We can see that
[tex]\displaystyle r+h=L[/tex]
Or, equivalently
[tex]\displaystyle h=L-r[/tex]
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:
[tex]\displaystyle r^2=h^2+\left(\frac{L}{2}\right)^2[/tex]
Replacing h
[tex]\displaystyle r^2=(L-r)^2+\left(\frac{L}{2}\right)^2[/tex]
Expanding squares
[tex]\displaystyle r^2=L^2-2Lr+r^2+\frac{L^2}{4}[/tex]
Simplifying
[tex]\displaystyle 2Lr=L^2+\frac{L^2}{4}[/tex]
Multiplying by 4
[tex]\displaystyle 8r=4L+L[/tex]
Joining terms
[tex]\displaystyle 8r=5L[/tex]
Solving for the ratio L/R as required
[tex]\displaystyle \frac{L}{r}=\frac{8}{5}[/tex]
The ratio of the side and the radius is 8:5
Since
r + h = L
So we can write h = L - r
Now we applied the Pythagoras theorem
[tex]r^2 = h^2 +( \frac{L}{2}) ^2\\\\r^2 = (L-r)^2 + ( \frac{L^}{2}) ^2\\\\r^2 = L^2- 2Lr+ r^2 + \frac{L^2}{4}\\\\ 2Lr = L^2 \frac{L^2}{4}\\\\[/tex]
Now
8r = 4L + L
8r = 5L
Learn more about the ratio here: https://brainly.com/question/13176103?referrer=searchResults
