To find the quadratic inequality that represents the length of the shorter leg \( x \), we first define the lengths of the legs of the right triangle.
Let:
- The shorter leg be \( x \) cm.
- The longer leg be \( x + 5 \) cm (since it is 5 cm longer than the shorter leg).
According to the Pythagorean theorem, the relationship between the lengths of the legs and the hypotenuse in a right triangle is given by:
\[ a^2 + b^2 = c^2 \]
Where:
- \( a \) and \( b \) are the lengths of the legs,
- \( c \) is the length of the hypotenuse.
In this case, we have:
\[ x^2 + (x + 5)^2 \geq 13^2 \]
Thus, we can expand and combine the expressions:
\[ x^2 + (x^2 + 10x + 25) \geq 169 \]
This simplifies to:
\[ 2x^2 + 10x + 25 \geq 169 \]
Rearranging gives:
\[ 2x^2 + 10x - 144 \geq 0 \]
Now, rearranging the terms, we rewrite this as:
\[ x^2 + 5x - 72 \geq 0 \]
However, the correct format of the inequality only represents the lengths derived from the Pythagorean theorem at a minimum hypotenuse of 13 cm.
Thus, the correct option that represents this inequality is:
\[ x^2 + (x + 5)^2 \geq 13^2 \]
So the correct choice is:
\( x^2 + (x + 5)^2 \geq 13^2 \)