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## High school geometry

### Course: High school geometry > Unit 8

Lesson 8: Inscribed shapes problem solving# Inscribed shapes: find diameter

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

## Want to join the conversation?

- Once you found out that the triangle is a right triangle, couldn't you use the Pythagorean triplet 8, 15, 17 to find the hypotenuse and diameter?(16 votes)
- That's exactly what Sal did. He determined it was a right triangle, then did
`8^2 + 15^2 = x^2`

and got`17`

.(28 votes)

- How did Sal know that angle C was a right angle?(8 votes)
- There is a theorem in geometry that says for any triangle with one side completely on the diameter of its circumscribed circle (the circle touching all three vertices of the triangle), then this triangle must be a right triangle, with the right angle where the two shorter lines of the triangle meets the circle.

It would be a good exercise for you to prove it, if you haven't already, as it's fairly easy to prove.(12 votes)

- Is the hypothenuse of an inscribed right triangle always a diameter?(4 votes)
- Yes, always. "If a right triangle is inscribed in a circle, then its hypotenuse is a diameter of the circle. If one side of a triangle inscribed in a circle is a diameter of the circle, then the triangle is a right triangle and the angle opposite the diameter is the right angle."

That quote is from here:

http://www.math.uakron.edu/amc/Geometry/HSGeometryLessons/InscribedRightTriangles.pdf(6 votes)

- Did anyone else do a completely different method to find that C was a right angle? I sort of forgot this was in the inscribed and circumscribed angle series, so I didn't do it that way.(4 votes)
- Which method did you use? I've been trying to come up with another one because of your comment but can't seem to figure it out :D.(2 votes)

- So how do we find angles when there are no side lengths?(3 votes)
- The only way to identify angles without side lengths that I know of is to use a protractor or means of indirect measurement, such as subtracting the two other angles from 180 degrees.(2 votes)

- i still do not get how < C is 90.care to explain?(2 votes)
- The central angle and the inscribed angle both subtend the same arc. since the central angle is 180 deg and the inscribed angle in 1/2 of than the measure of angle c was 90 deg.(3 votes)

- How did sal get 17? I understand the rest but 17 appeared and sal didn’t explain how to get 17.(2 votes)
- Sal got seventeen because the square root of 289 is 17, which means 17 times 17 equals 289. He got the square root in order to turn x^2 into x. He got 289 by using the pythagorean theorem since there is a right triangle that we have two side lengths for.(2 votes)

- prove why angle c is 90 degrees using statements and proof without the length of CB and AC and you had angle B being beta and angle A being alpha(2 votes)
- at around1:19i got lost, could someone explain it a little better please?(1 vote)
- Once Sal knows that the triangle is a right triangle, he applies the Pythagorean Therorm (a^2 + b^2 = c^2, where c is the length of the hypotenuse and a and b are the lengths of the other 2 sides)

So he squares each of the shorter sides (8 and 15), adds the results together (getting 289), and then takes the square root to find the length of the hypotenuse of the triangle (which is the diameter of the circle) as 17.(2 votes)

- At1:02, how can angle ACB is a right angle if we don't know if that angle is exactly 90 degrees? Like that could be an 89.7-degree angle.(1 vote)
- You know that if you have a diameter, it separates the circle into two 180 degree arcs. An inscribed angle is always 1/2 of the arc that it subtends. So 1/2*180=90.(2 votes)

## 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.