Bot GPT 4-o mini To find the length of the unknown side of a right triangle, we can use the Pythagorean theorem, which states that:
\[ a^2 + b^2 = c^2 \]
where \(c\) is the length of the hypotenuse, and \(a\) and \(b\) are the lengths of the other two sides.
In this case, you need to identify whether the side of length 14 is the hypotenuse or one of the other two sides. Let's consider two scenarios:
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If 14 is the hypotenuse: We can denote the unknown side as \(x\). So we have:
\[ x^2 + 7^2 = 14^2 \]
Simplifying:
\[ x^2 + 49 = 196 \]
\[ x^2 = 196 - 49 \]
\[ x^2 = 147 \]
\[ x = \sqrt{147} \approx 12.12 \]
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If 7 is the hypotenuse (though it doesn't typically make sense for a right triangle), we have:
\[ 14^2 + x^2 = 7^2 \]
This yields:
\[ 196 + x^2 = 49 \]
\[ x^2 = 49 - 196 \]
\[ x^2 = -147 \]
In this case, we see that it's impossible because you cannot have a negative value for \(x^2\).
Since the only valid scenario is when the hypotenuse is 14, the length of the unknown side is:
\[ x \approx 12.12 \]
The length of the unknown side is approximately 12.12 units.
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