Points $F$, $E$, and $D$ are on the sides $\overline{AB}$, $\overline{AC}$, and $\overline{BC}$, respectively, of right $\triangle ABC$ such that $AFDE$ is a square. If $AB = 12$ and $AC = 8$, then what is $AF$?

I find it funny how if you click AoPS and you see all of their posts, they're all related to "stop cheating!" requests.

please help fast

Aops has done this ALOT

the answer is 4, you are welcome

What is the answer though?

Ansa?

solve it yourself lol

To find the length of $AF$, we can use the Pythagorean Theorem to find the length of side $\overline{BC}$.

Let's start by labeling the triangle with the given information:

- $AB = 12$
- $AC = 8$

Using the Pythagorean Theorem, we have:

$(AC)^2 + (BC)^2 = (AB)^2$

$8^2 + (BC)^2 = 12^2$

$64 + (BC)^2 = 144$

Subtracting $64$ from both sides, we have:

$(BC)^2 = 80$

Now, we can find the length of side $\overline{BC}$:

$BC = \sqrt{80} = 4\sqrt{5}$

Since $AFDE$ is a square, $AF$ is equal to the length of $\overline{BC}$, so:

$AF = 4\sqrt{5}$

Therefore, $AF$ is $4\sqrt{5}$.

To summarize, to find the length of $AF$:
1. Use the Pythagorean Theorem to find the length of side $\overline{BC}$.
2. Since $AFDE$ is a square, $AF$ is equal to the length of $\overline{BC}$.
3. Simplify and find the length of $AF$ as $4\sqrt{5}$.

This is an Introduction to Geometry challenge problem.

#1: You're not supposed to be cheating on these challenge problems, or else they aren't challenging.

Subreason #1: This could get your account banned or suspended.

#2: The instructors in Introduction to Geometry are there for a reason.