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  • 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.

The center of this circle is O. And I

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B1 INT squared equal oa circle hypotenuse pythagorean theorem

Pythagorean theorem and radii of circles | Circles | Geometry | Khan Academy

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    onyi posted on 2016/02/21
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