To use the Pythagorean theorem, we must first determine whether the triangle is a right triangle. The theorem states that in a right triangle, the sum of the squares of the two legs equals the square of the hypotenuse.
Let's denote the sides of the triangle as follows:
- Side \( a = 17 \) cm (one leg)
- Side \( b = x \) cm (the other leg)
- Side \( c = 52 \) cm (the hypotenuse)
Given that \( c \) (52 cm) is longer than both \( a \) and \( b \), we can apply the Pythagorean theorem:
\[ a^2 + b^2 = c^2 \]
Substituting the known values:
\[ 17^2 + x^2 = 52^2 \]
Calculating the squares:
\[ 289 + x^2 = 2704 \]
Now, isolate \( x^2 \):
\[ x^2 = 2704 - 289 \]
\[ x^2 = 2415 \]
Taking the square root of both sides to solve for \( x \):
\[ x = \sqrt{2415} \]
Calculating the square root:
\[ x \approx 49.15 , \text{cm} \]
Thus, \( x \) is approximately \( 49.15 , \text{cm} \).