Bot GPT 4-o mini To solve this problem, we first define the legs of the right triangle. Let \( x \) be the length of the shorter leg. Since one leg is 5 centimeters longer than the other, the other leg can be described as \( x + 5 \).
According to the Pythagorean theorem, the relationship between the legs and the hypotenuse \( c \) of a right triangle is given by:
\[ x^2 + (x + 5)^2 = c^2 \]
In this case, we want the hypotenuse to be at least 13 centimeters long, which means:
\[ c \geq 13 \]
Thus, we need to ensure that:
\[ x^2 + (x + 5)^2 \geq 13^2 \]
Calculating \( 13^2 \):
\[ 13^2 = 169 \]
Substituting this into our inequality gives us:
\[ x^2 + (x + 5)^2 \geq 169 \]
Now we simplify \( (x + 5)^2 \):
\[ (x + 5)^2 = x^2 + 10x + 25 \]
Therefore, our inequality becomes:
\[ x^2 + x^2 + 10x + 25 \geq 169 \]
Combining like terms results in:
\[ 2x^2 + 10x + 25 \geq 169 \]
Subtracting 169 from both sides gives:
\[ 2x^2 + 10x - 144 \geq 0 \]
To match this with the original responses, we will analyze the provided options. The correct quadratic inequality representing the length of the shorter leg \( x \) is:
\[ x^2 + (x + 5)^2 \geq 13^2 \]
Thus, the correct response is:
\( x^2 + (x + 5)^2 \geq 13^2 \).
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