TITLE: Elementary Geometry
QUESTION [7 upvotes]: The side of the square measures $1\  \mathrm{cm}$ , and $AC = 1\  \mathrm{cm}$, find the value of $AB$

REPLY [2 votes]: Let D and E be the endpoints of the upper side of the square. We know that   $\triangle DEC \cong \triangle ABC. $ Therefore, we have the following: $$EC= \frac{1}{AB+1} ; 1=(1-EC)^2 + AB^2 $$ Solving for $AB$, we have $$ AB^4 + 2AB^3+AB^2-2AB-1 =0$$ Since we know that $0 \le AB \le 1,$ we apply the newton's method by letting $1$ be our initial value in the first iteration. Then we have: $$AB= 0.8832035059$$