express the hypotneuse as a function of a give (a+9) and (a)

Given the sides \( a+9 \) and \( a \) of a right triangle, we want to express the hypotenuse \( c \) as a function of \( a \).

Using the Pythagorean theorem:

\[
c = \sqrt{(a+9)^2 + a^2}
\]

Expanding the squares:

\[
(a+9)^2 = a^2 + 18a + 81
\]
\[
a^2 = a^2
\]

Adding these together:

\[
c = \sqrt{a^2 + 18a + 81 + a^2} = \sqrt{2a^2 + 18a + 81}
\]

So, the hypotenuse \( c \) as a function of \( a \) is:

\[
c(a) = \sqrt{2a^2 + 18a + 81}
\]

Would you like more details or have any questions?

Here are five related questions:
1. How would the function change if one of the sides was \( a-9 \) instead of \( a+9 \)?
2. Can you find the derivative of the hypotenuse function \( c(a) \) with respect to \( a \)?
3. What is the value of the hypotenuse when \( a = 5 \)?
4. How would you solve for \( a \) if the hypotenuse \( c \) is known?
5. How does the hypotenuse change as \( a \) increases?

**Tip:** When simplifying square roots, always check if the expression can be factored to make the process easier.

Learn how to express the hypotenuse of a right triangle as a function of the sides a+9 and a using the Pythagorean theorem. This problem involves calculating the hypotenuse based on given side lengths, suitable for advanced high school students.

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