If you're seeing this message, it means we're having trouble loading external resources on our website.

If you're behind a web filter, please make sure that the domains *.kastatic.org and *.kasandbox.org are unblocked.

Main content
Current time:0:00Total duration:8:19

Proof: radius is perpendicular to a chord it bisects

CCSS.Math:

Video transcript

in the last video we learned that if we have two different triangles and if all of the corresponding sides of the two triangles have the same length then by side-side-side we know that the two triangles are congruent and I also touched a little bit on the idea of an axiom or postulate but I want to be clear sometimes you will hear this referred to as a side side side theorem theorem and sometimes you'll hear it as a side-side-side postulate or axiom postulate or axiom and I think it's worth differentiating with these mean a postulate an axiom is something that you just assume you assume from the get-go while a theorem is something you prove using more basic or using some postulates or axioms so and really in all of mathematics you make some core assumptions you make some core assumptions you call these you call these the axioms or the postulates axioms axioms or the postulates and then using those you try to prove theorems so maybe using that one I can prove some theorems over here and maybe using that theorem and then this axiom I can prove another theorem over here and then using both of those theorems I can prove another theorem over here I think you get the picture this actually might lead us to this theorem these two might lead us to this theorem right over here and we essentially try to build our knowledge or rebuild a mathematics around these core assumptions in an introductory geometry class we kind of we don't rigorously prove side-side-side we don't rigorously prove the side-side-side theorem and that's why in a lot of geometry classes you kind of just take it as a given as a postulate or an axiom and the whole reason why I'm doing this is one just so you know the difference between the words theorem and postulate or axiom and also so that you don't get confused it is just a given but in a lot of books and I've looked at several books they do refer to it as the side-side-side theorem even though they never prove it rigorously they do just assume it so it really is more of a postulate or an axiom now with that out of the way we just take we're just going to assume going forward that we just know that this is true we're going to take it as a given I want to show you that we can already do something pretty useful with it so let's say that we have a circle let's say that we have a circle there's many useful things that we can already do with it and this circle has a center right here at a and let's say that we have a cord a cord in the circle that is not a diameter so let me draw a cord here so let me draw a cord in the circle so it's a kind of a segment of a secant line and let's say that I have let's say that I have a line that bisects that bisects this cord from from the center and I guess I could call it a radius because I'm gonna go from the center to to the edge of the circle right over there so I'm going to the center to the circle itself and when I say bisects it so these are all I'm just setting up the problem right now when I say bisecting it it means it splits that line segment in half so what it tells us is is that the length of this segment right over here is going to be equivalent to the length of this segment right over there and what I want to do is so this is I set it up I have a circle this radius bisects this cord right over here and what I want to do is prove the goal here is to prove is to prove that it bisects this cord at a right angle or another way to say it let me add some points here let's call this B let's call this C and let's call this D I want to prove that segment a B segment a B is perpendicular it intersects it at a right angle is perpendicular to segment CD to segment CD and as you can imagine I'm gonna prove it pretty much using the side-side-side whatever you want to call it side side side theorem postulate or axiom so let's do it let's think about it this way so you can imagine if I'm going to use this I need to have some triangles there's no triangles here right now but I can construct triangles that I can construct triangles based on things I know for example I can construct this has some radius so let's call this that's a radius right over here the length of that is just going to be the radius of the circle but I can also do it right over here the length of a see is also going to be the radius of the circle so we know that these two lines have the same length which is the radius of the circle which is a late radius of the circle we could say that ad is congruent to AC or they have the exact same lengths we know from the setup in the problem we know from the setup of the problem that this segment is equal in length to this segment over here we could even let me add a point here so I can refer to it so if I call that point E we know from the setup in the problem that C E is congruent to e d or they have the same lengths C e has the same length as IDI and we also know that both of these triangles the one here on the left and the one here on the right on the one here on the right they both share the side ei so EA is clearly equal to EA so this is clearly equal to itself it's the same side the same side is being used for both triangles the triangles are adjacent to each other and so we see a situation where we have a truck where we have two different triangles that have corresponding sides being equal this side is equivalent to this side right over here this side is is equal in length to that side over there and then we have obviously AE is equivalent to itself it's a side on both of them it's the corresponding side on both of these triangles and so by side-side-side so by side-side-side we know we know that triangle triangle ABC triangle a BC is congruent to triangle a II sorry it's not abc's AEC sorry we know she let me write it over here by side side side we know we know that triangle AEC AEC is congruent to triangle AED let's try and congruent to triangle a IDI but how does that help us how does that help us knowing that you know we used our little theorem but how does that actually help us here well what's cool is once we know that two triangles are congruent so because so because they are congruent that tells us so from that we can deduce that all the angles are the same and in particular we can deduce that this angle right over here that the measure of angle CEA see ei is equivalent to the measure of angle DEA d-ii a measure of angle DEA and the reason why that's all why that's useful is that we also see just by looking at this that they are supplementary to each other they're adjacent angles their outer sides form a straight angle so CEA is supplementary and equivalent to DEA so they are also supplementary so we also have the measure of angle CEA measure of angle CEA plus the measure of angle DEA is equal to 180 degrees but they're equivalent to each other so I could replace the measure of DEA with the measure of CEA measure of angle CEA or I could rewrite this as 2 times the measure of angle CEA is equal to 180 degrees or I could divide both sides by two and I say the measure of angle CEA is equal to 90 degrees which is going to be the same as the measure of angle DEA because they're equivalent so we know that this angle right over here is 90 degrees so I can do it with that little box and this angle right over here is 90 degrees and because a B intersects where it intersects CD we have a 90 degree angle here there and we could also prove this over there as well they are perpendicular to each other