If in a right​ triangle, the given legs are aequals18 and bequals27​, then by using the Pythagorean Theorem find the​ hypotenuse, c.
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Part 1
The hypotenuse is cequals
  
enter your response here.
​(Simplify your answer. Type an exact​ answer, using radicals as​ needed.)

To find the hypotenuse \( c \) of a right triangle with legs \( a = 18 \) and \( b = 27 \), we use the Pythagorean Theorem:

\[
c = \sqrt{a^2 + b^2}
\]

### Step-by-Step Solution

1. Substitute the values of \( a \) and \( b \):
   \[
   c = \sqrt{18^2 + 27^2}
   \]

2. Calculate \( 18^2 \) and \( 27^2 \):
   \[
   18^2 = 324
   \]
   \[
   27^2 = 729
   \]

3. Add these results:
   \[
   c = \sqrt{324 + 729} = \sqrt{1053}
   \]

4. Simplify \( \sqrt{1053} \) if possible.

The prime factorization of \( 1053 \) is \( 3^2 \times 3 \times 3 \times 13 = 9 \times 13 \times 9 \), which doesn’t yield a perfect square simplification.

Thus, the exact answer for the hypotenuse \( c \) is:
\[
c = \sqrt{1053}
\]

Would you like further details on simplifying radicals or any other part of this solution?

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Here are some related questions to explore:

1. What is the hypotenuse if both legs are multiples of a smaller right triangle?
2. How does changing one leg affect the hypotenuse in a right triangle?
3. Can you use the Pythagorean theorem to solve for missing leg values?
4. How do you verify if three given sides form a right triangle?
5. What is the relationship between the area of a right triangle and its legs?

**Tip:** For simplifying radicals, look for factors that are perfect squares, as they can be taken outside the radical.

If in a right triangle, the given legs are a=18 and b=27, then by using the Pythagorean Theorem find the hypotenuse, c.

Learn how to find the hypotenuse of a right triangle with legs a=18 and b=27 by applying the Pythagorean Theorem. Follow a step-by-step solution to calculate the exact answer for c using radicals. Suitable for students in Grades 8-10.

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