# Algebra Pythagorean Theorem

#### Ebba Sen Pai

Hello all. I was wondering how to solve this particular problem. I am not certain I am setting it up correctly, and I think it's likely I don't have it quite right. Any help is deeply appreciated.

(from textbook)
"Use the Pythagorean Theorem to find the missing side length. Leave answer in simplest form."

$$\displaystyle a= \frac{2}{3}b , c=12$$

(my attempt)

$$\displaystyle \left (\frac{2}{3}b \right )^2+b^2=12^2$$

Next I raise the contents of my first parenthesis to the second power, then combine like terms $$\displaystyle b^2$$.

Thus

$$\displaystyle \frac{4}{9}b +b^2$$

$$\displaystyle \frac{13}{9}b^2 =144$$
\Multiply both sides by reciprocal of b's coefficient
$$\displaystyle b^2=\frac{1296}{13}$$

My guess at the solution is the same as above, just apply radical to right and remove b's 2nd power.
I appreciate any and all input. My book provides no solution, so I don't know whether this is correct or miles off the mark.

Last edited by a moderator:

#### romsek

Math Team
This is correct though as they are lengths, and thus non-negative, you could go further
and say that

$b = \dfrac{36}{\sqrt{13}}=\dfrac{36\sqrt{13}}{13}$

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