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# pythagorean theorem

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The hypotenuse of right triangle $ABC$ is $AB$. The area of square $BCDE$ is $144$. If $AB$ and $AC$ are each integers less than $20$, how many possibilities are there for their lengths?

Aug 14, 2023

#1
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Hello,
I have solve the ans, you can read this

Let's analyze the problem step by step:

The hypotenuse of a right triangle is equal to one of its legs: $AB = AC$.
The area of square $BCDE$ is given as $144$. Since the square has sides equal to the legs of the right triangle, its area is also equal to the product of the legs: $AB \cdot AC = 144$.
We are asked to find integer values of $AB$ and $AC$ that are each less than $20$.
Now, we can find the possible integer pairs of $(AB, AC)$ that satisfy the conditions.

Since $AB = AC$, we have $AB^2 = 144$, which implies $AB = \sqrt{144} = 12$.

Therefore, there is only one possibility for the lengths of $AB$ and $AC$, which is $AB = AC = 12$.

So, there is only one possibility for the lengths of $AB$ and $AC$ that satisfies the given conditions.

I hope my ans. is good.

Aug 14, 2023
#2
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Let's analyze it step-by step:

- "The hypotenuse of a right triangle is equal to one of its legs: $AB = AC$."

That would make it degenerate

- "The area of square $BCDE$ is given as $144$. Since the square has sides equal to the legs of the right triangle, its area is also equal to the product of the legs: $AB \cdot AC = 144$."

It appears as if ChatGPT doesn't understand the difference between a hypotenuse and a leg. Furthermore, $$AC$$ is not part of the square.

- "Since $AB = AC$, we have $AB^2 = 144$, which implies $AB = \sqrt{144} = 12$."

I don't know where ChatGPT got this impression.

- "Therefore, there is only one possibility for the lengths of $AB$ and $AC$, which is $AB = AC = 12$.

So, there is only one possibility for the lengths of $AB$ and $AC$ that satisfies the given conditions."

I mean yeah, this logically follows from the previous steps which are all wrong. I'm pretty sure degenerate triangles don't count so the answer according to you(or ChatGPT should I say) is actually $$0$$.

- "I hope my ans. is good. "

It's not (good, nor is it yours).

plaintainmountain  Aug 14, 2023
edited by plaintainmountain  Aug 14, 2023
edited by plaintainmountain  Aug 14, 2023