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### Course: Geometry (all content)>Unit 14

Lesson 8: Inscribed shapes problem solving

# Proof: Right triangles inscribed in circles

Proof showing that a triangle inscribed in a circle having a diameter as one side is a right triangle. Created by Sal Khan.

## Want to join the conversation?

• I have a question Can I prove it this way too?

As Diameter is a line segment passing through the center and it has an angle of 180 degrees so the measure of the intercepted arc will be 180 degrees and then by the inscribed angle theorem that inscribed angle will be 90 degrees.
because inscribed angle = intercepted arc / 2
so the inscribed angle would be 180/2 = 90 degree.
• Yes; If two vertices (of a triangle inscribed within a circle) are opposite each other, they lie on the diameter. By the inscribed angle theorem, the angle opposite the arc determined by the diameter (whose measure is 180) has a measure of 90, making it a right triangle. Good job! :][
• At , why does sal use theta instead of a, b, c, etc?
• We use theta for angles in math. It is not so important now, but when you take trigonometry, you will use it all the time.
• What does Sal mean when he says subtends? As in, the inscribed angle subtends this arc.
• take an angle, put the vertex on the center point of a circle, and extend the rays of the angle to intersect the circumference of the circle. the arc in-between the 2 intersections on the circumference subtends the angle.❀
• So, say if I have a central angle Theta with measure of 24 degrees that subtends an arc. What is the relationship between the central angle and the subtended arc measure? Will the arc also be 24 degrees?. Now say I have an inscribed angle Psi that subtends the same arc. Will that inscribed angle be 12 degrees? Thanks in advance.
2) If Psi subtends the same arc as Theta, then yes, Theta = 2Psi
1) The relationship between a central angle Theta and the subtended angle is proportionate to the circumference and 360 degrees like so:
(Theta)/(360) = (Arc length)/(circumference)
• Couldn't we demonstrate in a more simplified way?

Given that central angle / 2 = inscribed angle

The angle of the diameter (180 °) is the central angle that subtends the arc represented by half the circumference.

Tracing a triangle with the diameter being one of the sides, we would automatically form an inscribed angle that also subtends the same arc as the angle of the diameter.

Thus, that inscribed angle would be half of 180 ° (90 °), that is, a right angle
• Hi Levi, the answer is, YES! That is a great proof. There doesn't always have to be only one proof for a theorem, so knowing and understanding different ideas and concepts that build different proofs is always good.
Sorry for the late reply, but I hope that helped!
~Hannah
• Why does Sal use theta? Why not beta or gamma?
• Classically, lowercase theta is used to describe an unnamed angle. Interesting, fact, the archaic form, a symbol of a plus sing in a circle also represents Earth.
• I don't think this video described what an Inscribed Angle is at all.
• An inscribed angle in a circle is an angle with the vertex on the circle, and its sides are chords in that circle
• Wouldn't that make all triangles right triangles? Because you could draw a circle around the triangle with the longest side being the diameter, and according to this proof, the opposite angle would be right? Or am I going crazy?
• No, this only proves that the triangle is a right triangle if and only if one of the sides is a diameter. There are other triangles that can be drawn that are inscribed in circles that are not right triangles. If you draw a circle and put any three points such that any two points do not form a diameter, then the triangle will not be a right triangle.
• I have a question. A regular hexagon with area 54√3 (inch square) is inscribed in a circle. What is the circumference of the circle? Express answer in terms of π.
• A regular hexagon can be be cut into 6 equilateral triangles. In your case, each triangle will have area 54√3/6=9√3. The area of an equilateral triangle with side length s is s²√3/4. Since we know the areas of these triangles, we can solve for their side lengths:
s²√3/4=9√3
s²/4=9
s²=36
s=6

So the triangles have sides of length 6. And when follow a diameter of the circumcircle, we trace two sides of equilateral triangles. So the circle has diameter 6·2=12, and radius 6. So its circumference is 2·π·6=12π.

If your circle had circumference 25π, it would have area of roughly 490. 54√3 is roughly 93, so the inscribed hexagon would have to occupy just under 1/5th of the circles area, which doesn't sound reasonable.
• So, any inscribed angle opposite of a diameter (whose corresponding arc is 180), equals 90 because an inscribed angle = 1/2 the measure of its arc right?
• You are correct. Any inscribed angle that subtends a diameter is equal to 90 degrees. :)

## Video transcript

Let's say we have a circle, and then we have a diameter of the circle. Let me draw my best diameter. That's pretty good. This right here is the diameter of the circle or it's a diameter of the circle. That's a diameter. Let's say I have a triangle where the diameter is one side of the triangle, and the angle opposite that side, it's vertex, sits some place on the circumference. So, let's say, the angle or the angle opposite of this diameter sits on that circumference. So the triangle looks like this. The triangle looks like that. What I'm going to show you in this video is that this triangle is going to be a right triangle. The 90 degree side is going to be the side that is opposite this diameter. I don't want to label it just yet because that would ruin the fun of the proof. Now let's see what we can do to show this. Well, we have in our tool kit the notion of an inscribed angle, it's relation to a central angle that subtends the same arc. So let's look at that. So let's say that this is an inscribed angle right here. Let's call this theta. Now let's say that that's the center of my circle right there. Then this angle right here would be a central angle. Let me draw another triangle right here, another line right there. This is a central angle right here. This is a radius. This is the same radius -- actually this distance is the same. But we've learned several videos ago that look, this angle, this inscribed angle, it subtends this arc up here. The central angle that subtends that same arc is going to be twice this angle. We proved that several videos ago. So this is going to be 2theta. It's the central angle subtending the same arc. Now, this triangle right here, this one right here, this is an isosceles triangle. I could rotate it and draw it like this. If I flipped it over it would look like that, that, and then the green side would be down like that. And both of these sides are of length r. This top angle is 2theta. So all I did is I took it and I rotated it around to draw it for you this way. This side is that side right there. Since its two sides are equal, this is isosceles, so these to base angles must be the same. That and that must be the same, or if I were to draw it up here, that and that must be the exact same base angle. Now let me see, I already used theta, maybe I'll use x for these angles. So this has to be x, and that has to be x. So what is x going to be equal to? Well, x plus x plus 2theta have to equal 180 degrees. They're all in the same triangle. So let me write that down. We get x plus x plus 2theta, all have to be equal to 180 degrees, or we get 2x plus 2theta is equal to 180 degrees, or we get 2x is equal to 180 minus 2theta. Divide both sides by 2, you get x is equal to 90 minus theta. So x is equal to 90 minus theta. Now let's see what else we could do with this. Well we could look at this triangle right here. This triangle, this side over here also has this distance right here is also a radius of the circle. This distance over here we've already labeled it, is a radius of a circle. So once again, this is also an isosceles triangle. These two sides are equal, so these two base angles have to be equal. So if this is theta, this is also going to be equal to theta. And actually, we use that information, we use to actually show that first result about inscribed angles and the relation between them and central angles subtending the same arc. So if this is theta, that's theta because this is an isosceles triangle. So what is this whole angle over here? Well it's going to be theta plus 90 minus theta. That angle right there's going to be theta plus 90 minus theta. Well, the thetas cancel out. So no matter what, as long as one side of my triangle is the diameter, and then the angle or the vertex of the angle opposite sits opposite of that side, sits on the circumference, then this angle right here is going to be a right angle, and this is going to be a right triangle. So if I just were to draw something random like this -- if I were to just take a point right there, like that, and draw it just like that, this is a right angle. If I were to draw something like that and go out like that, this is a right angle. For any of these I could do this exact same proof. And in fact, the way I drew it right here, I kept it very general so it would apply to any of these triangles.