To determine the hypotenuse of a triangle with sides 12, 16, and 20, we can apply the Pythagorean theorem, which states that in a right 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 other two sides.
Let’s identify the longest side, which in this case is 20. We need to check if the sides satisfy the Pythagorean theorem:
\[ a^2 + b^2 = c^2 \]
where \( c \) is the hypotenuse.
Here, let's take \( a = 12 \), \( b = 16 \), and \( c = 20 \):
\[ 12^2 + 16^2 = 20^2 \]
Calculating each side:
\[ 144 + 256 = 400 \]
Simplifying the left side:
\[ 400 = 400 \]
Since the equation holds true, this confirms that the triangle with sides 12, 16, and 20 is a right triangle, and indeed the hypotenuse is 20.
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