In a right angled triangle, with angle at A being , the side $AB$ is of length 4cm and $BC$ is 15 cm. What is the length of side $AC$? 
To find the length of side \( AC \) in the right-angled triangle \( \triangle ABC \), we can use the Pythagorean theorem. The Pythagorean theorem states that in a right-angled triangle, the square of the length of the hypotenuse (the side opposite the right angle) is equal to the sum of the squares of the lengths of the other two sides. The theorem is given by:
\(c^2 = a^2 + b^2\)
Here, \( BC \) is the hypotenuse, \( AB = 4 \text{ cm} \), and \( BC = 15 \text{ cm} \). We need to find \( AC \).
According to the Pythagorean theorem:
\(BC^2 = AB^2 + AC^2\)
Plugging in the known values:
\((15)^2 = (4)^2 + AC^2\)
\(225 = 16 + AC^2\)
Subtracting 16 from both sides:
\(225 - 16 = AC^2\)
\(209 = AC^2\)
Taking the square root of both sides gives:
\(AC = \sqrt{209}\)
Hence, the length of side \( AC \) is \(\sqrt{209}\text{ cm}\). Therefore, the correct answer is option \(\sqrt{209}\text{ cm}\).
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