 In this video we want to prove that Equal chords of the same circle are equidistant from the center and here We have a diagram where we have two chords a b and pq and the center is o and we have drawn perpendicular from o to pq and a b and why have we drawn these perpendicular is Because we know that the distance or the shortest distance between a chord and a point or a line and a point The perpendicular distance and that's why we have drawn the perpendicular distance. What we need to prove here is that if Pq is equal to a b then o m is Equal to o c right, so this is what we need to prove How are we going to do this is first of all trying to collect all the information that we have What I can see from the diagram is that clearly pq is Equal to a b because it's given to us We are going to use a knowledge point that the perpendicular drawn from the center of a circle to the chord Biceps the chord so from our knowledge of geometry We know that m is a midpoint of pq because o m is a perpendicular from the center of the circle to the chord and similarly C is the midpoint of a b since o c is the perpendicular to a b and No is a center of the circle and with all these knowledge points. Let's see if we can prove O m is equal to o c now Let us first of all try and complete a triangle and why I like to draw the triangles in such Circle proves because if I can complete a triangle where o m and o c are the sides of those triangles Then it becomes easier for me to prove that the triangles are congruent And then I can show that o m and o c are congruent and because I can see different elements that are already equal I think this approach of making a triangle first and trying to show them congruent is going to work So we have completed two triangles one of them is triangle o m q and The other triangle is o c b So we mentioned in triangle o m q and triangle o c b what we see is that there are two right angles One of them is o m q major angle o m q is equal to major angle o c b And both are equal to 90 degrees And we know this from construction Right because we already know that these are perpendicular. So the angle is 90 degrees So can we say that the triangles that we're considering are Right-angled triangles Now we go ahead We have already mentioned one of the points the next point that we can consider in both of these triangles is side o q Right o q Is equal to side o b and why is that this is because both are Red eye of same circle and we already know that the red eye of the same circle are equal So we already got the second point in the two considered triangles Now, let's see if we can find something else We already know that p q is equal to a b right and therefore The half of both of them should be equal as well We already know m is the midpoint of p q and c is the midpoint of a b and therefore Can I say p m is equal to m q and because both of these chords are equal I can also say that this is also equal to ac and b c And we only want to highlight m q is equal to b c right and so We will just keep m q and b c here, right? Let me just write it well. So m q Is equal to b c and I can say this is deduced knowledge from the fact a perpendicular drawn from Center to the chord bisects it you won't really have to write Such a long reason in the exam, but I'm just writing it for you to understand This is a deduced knowledge from the given point as well as something that you know from the geometry So we have got three elements in the two triangles that we considered to be congruent or to be equal And it is enough to show that the two triangles triangle o m q is congruent with triangle O c b and which test that we used here We showed that these both are right angle triangles and the hypotenuse o q and o b are equal, right? So this was the hypotenuse, right? And we already showed One other element of both the triangles to be equal as well, which is m q equal to b c and therefore this is hypotenuse side test and since triangle o m q and triangle o c b are equal All the other corresponding elements of both the triangles are going to be equal and therefore o m corresponds to o c I can say that o c Is equal to o m and this is exactly what we wanted to prove at the top and therefore The equal chords of the same circle are equidistant from it