Current time:0:00Total duration:5:53

0 energy points

# Pythagorean theorem & radii of circles

Video transcript

The center of this
circle is O. And I apologize if I'm a
little out of breath, I actually just
did some pull-ups. Anyway, the center
of this circle is O. Find the exact
length of OA, CD, and OF. So let's look at each of these. So CD, it's part of
a right triangle. It's one of the two shorter
sides of a right triangle. But we don't know
the hypotenuse, so we're not going to
be able to figure out CD out right, just yet. Same thing with OF. OF is one of the two shorter
sides of a right triangle. But we don't know its
hypotenuse either. Now let's look at OA. OA is the hypotenuse
of a right triangle, and they've given us
the two other sides. So we can use the
Pythagorean theorem to figure out the hypotenuse. So we know that 7 squared,
let's call this x. We know that 7 squared
plus 24 squared is going to be equal to
the length of OA squared. It's going to be
equal to x squared. 7 squared is 49,
and 24 squared, well let's do a multiplication
right over here to figure out 24 times 24. 4 times 4 is 16. 4 times 2 is 8
plus-- so that's 96. And then two times 24 is 48. Add them together, we get 6. 9 plus 8 is 17, 576. So 49 plus 576 is
equal to x squared. And so let's think about
what this is going to be. And this is going to be the
same thing as 50 plus 575. I just took one away from
this and added one here. So 50 plus 575 is 625. So 625 is equal to x squared. And you might recognize
that 25 times 25 is 625. So x is equal to 25. And if you don't believe me
you could multiply that out on your own. So x is equal to 25. Or another way of thinking
about it, the exact length of OA is equal to 25. Now, how can we somehow
use that information to figure out this other stuff? Well all of these
other right triangles, all of their hypotenuses
are a radius of the circle, and so is OA. OA is a radius of the circle. OG is a radius of the circle. OC is a radius of the circle. Well, we just figured out the
radius of the circle is 25. So OG is going to be 25, and
OC is going to be 25 as well. So now we just have to apply the
Pythagorean theorem a few more times. So right over
here, if I call OF, let's just call that, I don't
know for the sake of argument, let's call that
length equal to a. So here, for this
triangle, we see that a squared plus
the square root of 141 squared-- I'll just write that
as 141-- so plus 141 is going to be equal to 25 squared,
which we already know to be 625. If we subtract 141
from both sides let's see where do we get. So let's do 625 minus 141
we get 5 minus 1 is 4. And then 12 here, and
we can put a 5 there. 12 minus 4 is 8. 5 minus 1 is 4. So we get 484. So we get a squared
is equal to 484. So what squared is equal to 484? Actually, I'll just try to
do a prime factorization here to figure this out. So 484 is 2 times 242,
which is 2 times 121, which is the same thing
as 11 times 11. So another way of
thinking about it is this is 2 squared-- so 484,
I'll write it over here. 484 is equal to 2
squared times 11 squared. Or it's the same
thing as 2 times 11 squared, which is 22 squared. So in this case a is equal to--
let me just clean all that up so I have some space to work
with-- a is equal to 22. Let me write that down. a is equal to 22, so that's
equal to 22 right here. And that's the
length of segment OF. So this is 22. And then finally CD,
once again we just apply the Pythagorean theorem. Let's just call this, I don't
know, I've already used a. I've already used x. I don't know, I'll call this b. So we see that b squared
plus 15 squared, which is the same thing as
225-- 15 squared is 225-- is going to be
equal to 25 squared, is going to be equal to 625. Subtract 225 from both sides you
get b squared is equal to 400, and the square root of 400
is pretty easy to calculate. B is equal to 20. So segment OA is 25,
CD is 20, and OF is 22.