Question

Question is attached. I would like an explanation of how to do the problem as well as an answer please. :)

Answer 1

Ok, so we have a bunch of right triangles. Probably the most useful bit of info will be the Pythagorean theorem. \[a^2 +b^2 = c^2\]So looking at the large triangle we have:
\[x^2 + y^2 = (3+9)^2 = 12^2 = 144\]

Looking at the smaller triangles we see that they share a side.. Lets call the length of that side \(S\).

So we have two equations:

\[9^2 + S^2 = y^2\]\[3^2 + S^2 = x^2 \]
Putting the three together we get
\[x^2 + y^2 = 144\]\[\implies (3^2 + S^2) + (9^2 + S^2) = 144\]

And from there you just solve for \(S^2\) and plug back in to find \(y^2\)

Answer 2

I don't understand how to solve for s^2 if 3^2 + 9^2 is 144.

Answer 3

\[(3^2 + S^2) + (9^2 + S^2) = 144\]\[\implies 9 + S^2 + 81 + S^2 = 144\]\[\implies 2S^2 = 144 - 90\]\[\implies S^2 = \frac{54}{2} = 27\]

\[y^2 = 9^2 + S^2 \]\[\implies y^2 = 81 + 27 = 108\]\[ \implies y = 2\sqrt{27}\]