Question

The circle with center A and radius AB is inscribed in the square here. AB is extended to C. What is the ratio of AB to AC?

Answer 1

let me add a few things to your drawing:
since AB is radius of the circle, we will call it R

now look at the triangle ACD.
it has 3 sides: AC, AD, CD
and the angle at point B is 90 degrees, hence we can apply the Pythagorean theorem
\[c ^{2} = a ^{2} + b ^{2}\]\[AC ^{2} = AD ^{2} + CD ^{2}\]
from the given picture we see that AB, AD and CD are all the same length, length of radius of given circle, hence all of them = R
\[AC ^{2} = R ^{2} + R ^{2}\]\[AC ^{2} = 2*R ^{2}\]
we square all of it and get:
\[AC = \sqrt{2}*R\]

and now, you are looking for AB/AC. can you tell me what is the answer?

Answer 2

OH NO!