It's a right angle triangle and you know one of the sides is 5 cm the other is 3 cm what is the third side

Step 1: Identify the given information

We are given a right-angled triangle with two side lengths: $3\text{ cm}$ and $5\text{ cm}$. We need to determine if these are the two legs (perpendicular sides) or one leg and the hypotenuse.

Step 2: Apply the Pythagorean Theorem

The Pythagorean Theorem states that in a right-angled triangle, the square of the hypotenuse (the side opposite the right angle) is equal to the sum of the squares of the other two sides (legs). Here, $a$ and $b$ are the lengths of the legs, and $c$ is the length of the hypotenuse.

Step 3: Case 1: Given sides are legs

If the given sides, $3\text{ cm}$ and $5\text{ cm}$, are the legs of the right-angled triangle, then we can find the hypotenuse $c$ using the Pythagorean theorem. We calculate $3^2 + 5^2 = 9 + 25 = 34$. So, $c^2 = 34$, which means $c = \sqrt{34}\text{ cm}$.

Step 4: Case 2: Given sides are one leg and the hypotenuse

If $5\text{ cm}$ is the hypotenuse and $3\text{ cm}$ is one of the legs, we can find the other leg, $a$. We set up the equation $a^2 + 3^2 = 5^2$. This simplifies to $a^2 + 9 = 25$, so $a^2 = 16$. Taking the square root, we find $a = 4\text{ cm}$. This is a common Pythagorean triplet (3, 4, 5).

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