To find the hypotenuse of a right triangle using the Pythagorean theorem, we follow these steps:
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Identify the theorem: The Pythagorean theorem states that in a right triangle, the square of the length of the hypotenuse (c) is equal to the sum of the squares of the lengths of the other two sides (a and b). This can be written as: \[ c^2 = a^2 + b^2 \]
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Substitute the known values: If we know the lengths of the two legs of the triangle (let’s say leg a = 3 and leg b = 4), we plug these values into the equation: \[ c^2 = 3^2 + 4^2 \]
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Calculate the squares: Compute the squares of the legs: \[ c^2 = 9 + 16 \]
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Add the squares together: \[ c^2 = 25 \]
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Take the square root to find the hypotenuse: \[ c = \sqrt{25} = 5 \]
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Approximation of non-perfect square roots: If instead we had values that do not yield a perfect square (e.g., a = 2 and b = 3), we would follow the same steps: \[ c^2 = 2^2 + 3^2 = 4 + 9 = 13 \] Then, \[ c = \sqrt{13} \approx 3.61 \] (using a calculator or estimating that it lies between 3.6 and 3.7)
Summary:
To find the hypotenuse \( c \) of a right triangle with legs of lengths \( a \) and \( b \), apply the Pythagorean theorem: \[ c = \sqrt{a^2 + b^2} \] Calculate \( c \) based on given leg lengths, and if \( c \) results in a non-perfect square, approximate the square root for a practical value.