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### Course: High school geometry>Unit 8

Lesson 9: Proofs with inscribed shapes

# 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)
• 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.
• 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
• 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
• 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.