Proof: Right triangles inscribed in circles (video) | Khan Academy
Question: Prove that if you draw a triangle inside a semicircle, the angle opposite the diameter Draw a radius 'r' from the (right) angle point C to the middle M. The angle inscribed in a semicircle is always a right angle (90°). Try this Drag any orange The triangle formed by the diameter and the inscribed angle ( triangle ABC above) is always a right triangle. Relationship to Thales' Theorem. This is a. A right triangle (American English) or right-angled triangle (British English) is a triangle in which The relation between the sides and angles of a right triangle is the basis for trigonometry. The side .. The triangle can be inscribed in a semicircle, with one side coinciding with the entirety of the diameter (Thales' theorem).
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 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 degrees, or we get 2x plus 2theta is equal to degrees, or we get 2x is equal to 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.
Proof that the angle of a triangle in a semicircle is 90°.
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.
Proof that the angle of a triangle in a semicircle is 90°
The Inscribed Angle Conjecture I gives the relationship between the measures of an inscribed angle and the intercepted arc angle. It says that the measure of the intercepted arc is twice that of the inscribed angle. The precise statements of the conjectures are given below. Each conjecture has a linked Sketch Pad demonstration to illustrate its truth proof by Geometer's Sketch Pad!
The linked activities sheet also include directions for further "hands on" investigations involving these conjectures, as well as geometric problems which utilize their results. The precise statement of the conjectures: Conjecture Inscribed Angles Conjecture I: In a circle, the measure of an inscribed angle is half the measure of the central angle with the same intercepted arc.
In a circle, two inscribed angles with the same intercepted arc are congruent. The measure of each inscribed angle is exactly half the measure of its intercepted arc. Since they have the same intercepted arc, they have the same measure. Any angle inscribed in a semi-circle is a right angle. The intercepted arc for an angle inscribed in a semi-circle is degrees.