To solve for \( x \) in a triangle with sides of lengths 25 cm, 100 cm, and \( x \) using the Pythagorean Theorem, we first need to determine if this is a right triangle. The Pythagorean Theorem states that in a right triangle:
\[ a^2 + b^2 = c^2 \]
where \( c \) is the length of the hypotenuse (the longest side), and \( a \) and \( b \) are the lengths of the other two sides.
In this case, we can assume \( 100 \) cm is the longest side (hypotenuse). Therefore, we can set up the equation as follows:
\[ 25^2 + x^2 = 100^2 \]
Calculating the squares:
\[ 625 + x^2 = 10000 \]
Now, we can isolate \( x^2 \):
\[ x^2 = 10000 - 625 \] \[ x^2 = 9375 \]
Next, we take the square root of both sides to find \( x \):
\[ x = \sqrt{9375} \] \[ x \approx 96.19 \text{ cm} \]
Therefore, the length of the third side \( x \) is approximately \( 96.19 \) cm.
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