about 2 400 years ago there lived a man known as euclid he was a greek mathematician and he is known to be the father of geometry and that is why we typically call this type of geometry euclidean geometry so imagine euclid was playing around with a few lines and circles back in the day and what he discovered was because well because no one had really done this prior to him and he discovered a few interesting things but these things that he's discovered as we are going to see in this video they're not very difficult to understand but what he did is he he came up with all of the theories and summarized all of them so that we can use them without having to go discover them ourselves but let's take a look so the first thing he discovered was the following so i want you to if possible follow along with me and draw the circles and all of that so we're going to draw a circle first then what we're going to do is locate the center point and it doesn't have to be a hundred percent accurate we're just trying to get an idea of what euclid was trying to say next we're going to draw a chord now a chord is a line that touches two sides of the circle but it mustn't go through the center okay so something like this now what we're going to do is draw a line from the center to the halfway point of that chord and so because we know that it's at the halfway point we can quickly say that this line is the same as that line because we said it's the halfway point okay now what euclid then discovered when he was doing that was that this angle over here always ends up being 90 degrees and you can try that as many times as you like you will always find that if your line goes from the center and it hits another chord any chord in in half so it must be in half then it will form 90 degrees and so that was one of euclid's first circle geometry theorems so he said that this angle will always be 90 degrees as long as the line from the center goes to the midpoint so we can say that angle obc for example will equal 90 degrees and his reason was that we have a line from the center oh and i must just stop on the word center in different parts of the world people spell center in different ways and i personally just prefer to spell center like that whereas in other parts of the world and maybe you prefer to do it like this you do it with the er okay so line from center to the midpoint meaning the halfway point of a chord there we go so the reason that that is 90 degrees is because the line that comes from the center to the midpoint of the chord that's all you have to say for that one and we could very easily show how this wouldn't work if the line wasn't to the midpoint for example if we did this so that line that i've now drawn from the center is not going to the midpoint of the chord and so clearly we can see that this angle over here will not be 90 degrees so it has to go to the midpoint now we can switch things up in the opposite way so let's go and draw a line from the center and try and make it form an angle of 90 degrees with the chord it doesn't have to hit halfway i just want you to make it be 90 degrees like that over there now we're not going to say midpoint or we're not going to show that because that's not necessarily true what we wanted to happen was we made sure that that was 90 degrees and what euclid found was he took a ruler and he measured this length and then he measured this length over here so from here up to here and he found that those lengths were equal and so he put two lines like that and so we can say that if you have a line coming from the center and it hits the chord at 90 degrees then it will automatically divide that chord into two equal halves so mathematically we can say that a b is equal to bc and the reason for this one is because we have a line from the center not to the midpoint this time but making an angle of 90 degrees with the chord so we show it like that line from center perpendicular meaning 90 degrees to the chord okay so i hope you can see the slight difference between those two theorems and so what could we say with the circle over here well we've got the 90 degrees so because it's from the center we can say that ac is equal to bc so ac is equal to bc and the reason is is that we have a line from center now are we going to say mid to the midpoint of the chord or perpendicular to the chord you have to look at what you had first well what we had first was the 90 degrees so we're going to say perpendicular to chord so what it says is that these ac is the same as bc why because of the fact that we have the line from the center perpendicular to the chord with this one over here well now we've been given that bc is the same length as ac we can see that by those lines and so we can say that this angle must be 90 because it comes from the center and so we can say that angle o c a should equal 90 degrees y now it's because we have the line from the center to the midpoint because that's what we had first of chord so now we won't say because it's perpendicular we'll say it's because because of the fact that we had the two midpoints or we or c was at the midpoint we can then say because of that this angle must be 90 degrees so go ahead practice this a few times do it on the on your exam pad and you'll see that euclid is perfectly correct in what he discovered it always works every single time