Bot GPT 4-o mini Ariel's work contains a mistake in the equations she wrote to determine if the triangle with side lengths 13, 12, and 5 forms a right triangle. To assess whether these sides form a right triangle, we should apply the Pythagorean theorem, which states that in a right triangle, the squares of the lengths of the two shorter sides (the legs) should add up to the square of the length of the longest side (the hypotenuse).
In this case, the longest side is 13, and the two shorter sides are 12 and 5. According to the Pythagorean theorem, we need to check if:
\[ a^2 + b^2 = c^2 \]
where \( a = 5 \), \( b = 12 \), and \( c = 13 \):
Calculating the squares:
- \( 5^2 = 25 \)
- \( 12^2 = 144 \)
- \( 13^2 = 169 \)
Now, summing the squares of the shorter sides:
\[ 5^2 + 12^2 = 25 + 144 = 169 \]
Since \( 169 \) is equal to \( 13^2 \), we have:
\[ 5^2 + 12^2 = 13^2 \]
This means the triangle with side lengths 13, 12, and 5 does indeed form a right triangle. Therefore, Ariel's conclusion that these side lengths do not form a right triangle is incorrect. The correct conclusion is that these sides do form a right triangle based on the Pythagorean theorem.
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