About this video
This video finds the diameter of a circle with a triangle ACB inscribed so that AB is a diameter. Because inscribed angle C intercepts a semicircle (180 degrees), it is a right angle, so the Pythagorean theorem gives 15^2 + 8^2 = 289 = 17^2 and the diameter is 17.
You'll be able to
- explain why an inscribed angle that intercepts a semicircle is a right angle.
- recognize that the diameter is the hypotenuse of such a right triangle.
- use the Pythagorean theorem to find the diameter.
Key words
- diameter
- A chord that passes through the center of the circle; it is twice the radius.
- inscribed angle
- An angle whose vertex sits on the circle.
- hypotenuse
- The longest side of a right triangle, opposite the right angle.
- Pythagorean theorem
- In a right triangle, a^2 + b^2 = c^2, where c is the hypotenuse.
Transcript
What I want to do in this video is attempt to find the diameter of this circle right over here. And I encourage you to pause the video and try this out on your own. Well, let's think about what's going on right over here. AB is definitely a diameter of the circle.
It's a straight line. It's going through the center of the circle. O is the center of the circle right over here. And so what do we know? Well, we could look at this angle right over here, angle C, and think about it is an inscribed angle. And think about the arc that it intercepts.
It intercepts this arc right over here. This arc is exactly half of the circle. Angle C is inscribed. If you take these two sides or the two sides of the angle, it intercepts at A and B, and so it intercepts an arc, this green arc right over here. So the central angle right over here is 180 degrees, and the inscribed angle is going to be half of that.
It's going to be 90 degrees. Or another way of thinking about it, it's going to be a right angle. And what that does for us is it tells us that triangle ACB is a right triangle. This is a right triangle, and the diameter is its hypotenuse. So we can just apply the Pythagorean theorem here.
15 squared plus 8 squared-- let me do this in magenta-- is going to be the length of side AB squared. So this side right over here, let me just call that x. That's going to be equal to x squared. So 15 squared, that's 225. 8 squared is 64, plus 64-- I want to do that in green-- is equal to x squared.
225 plus 64 is 289 is equal to x squared. And then 289 is 17 squared. And you could try out a few numbers if you're unsure about that. So x is equal to 17. So the diameter of this circle right over here is 17.
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