Use the Pythagorean theorem to find the unknown side of the right triangle.

9
12

A right triangle has a vertical side of length 9, a horizontal side of length 12, and an unlabeled hypotenuse.
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Part 1
Hypotenuse lengthequals
  
StartRoot 225 EndRoot
​(Simplify your answer. Type exact​ answers, using radicals as​ needed.)

1 answer

To find the length of the hypotenuse of a right triangle using the Pythagorean theorem, you use the formula:

\[ c^2 = a^2 + b^2 \]

where \(c\) is the length of the hypotenuse, and \(a\) and \(b\) are the lengths of the other two sides. In this case:

  • \(a = 9\)
  • \(b = 12\)

Now, plug in the values:

\[ c^2 = 9^2 + 12^2 \]

Calculating the squares:

\[ c^2 = 81 + 144 \]

Adding those together:

\[ c^2 = 225 \]

Now, take the square root to find \(c\):

\[ c = \sqrt{225} \] \[ c = 15 \]

Thus, the length of the hypotenuse is \(15\). However, it seems you've provided a different context by stating "StartRoot 225 EndRoot."

To give you an exact answer in the requested format:

\[ \text{Hypotenuse length} = 15 \]

and in radical form:

\[ = \sqrt{225} \]

So the simplified exact answer is either \(15\) or \( \sqrt{225} \).