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# Proof: Right triangles inscribed in circles

CCSS.Math:

## Video transcript

let's say we have a circle and then we have a diameter of this circle let me drew my best draw my best diameter that's pretty good this right here is the diameter of the circle er it's a diameter of the circle that's the diameter and let's say I have a triangle where the diameter is one side of the triangle and the angle opposite that side its vertex sits someplace 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 is going to be a right triangle and the 90 degrees 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 if we can what we can do to show this well we have in our toolkit the notion of an inscribed angle its relation to a central angle that subtends the same arc so let's look at that so let's say that this is it this is an inscribed angle right here let's call this theta that angle theta and I'll let me so 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 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 it subtends that arc up there the central angle that subtends that same arc is going to be twice this angle we prove that several videos ago so this is going to be two theta 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 rotate it I could draw it like this I flipped it over it would look like that that then the green side would be down like that and both of these sides are of length are this top angle is to theta 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 it's two sides or equal this is isosceles so these two base angles must be the same so these two 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 2 theta have to equal 180 degrees they're all in the same triangle so let me write that down we get X plus X plus 2 theta all have to be equal to 180 degrees or we get 2x plus 2 theta is equal to 180 degrees or we get 2x is equal to 180 minus 2 theta divide both sides by 2 you get X is equal to 90 minus theta so X is equal to X is equal to 90 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 that to actually show that first result about about inscribed angles and the relation between them and center angle subtending the same mark so this is data that's data because this is an isosceles triangle so what is this whole angle over here what is that whole angle over here oh it's going to be theta plus 90 minus theta that angle right there is 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 a diameter and then the angle or the vertex of the angle opposite sits opposite of that side sits on the circumference then on then this angle right here is going to be is going to be a right angle and this is going to be a right triangle so if I just want 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 would do 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