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# Inscribed shapes: find diameter

Find the diameter of a circle using an inscribed right triangle. Created by Sal Khan.

Video 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.