The circle geometry problem most students get wrong.

Let's find the radius of this circle. Yes, I know where your mind went but we have two different methods we can use to achieve that. Welcome back to my channel. It's been a while and I'm happy to shoot this video. So let's get straight to the point. So we are asked to find the radius First of all, let's call this point A, point B, point O and point C. Now I want to extend line AO to the circumference of the circle. Please note that this is not drawn to scale. So I want to extend AO to the circumference of the circle. Yes, that is because I want to make use of the chord theorem, right? Now the chord theorem states that when two chords intersect inside the circle, right? When two chords intersect each other inside the circle. Now the product of the segments of the first chord is equal to the product of the segment of the other chord, okay? So let's assume this is A, this is B, this is C, and this is D. Now let's call this capital AB, okay? And call this chord capital CD. Now AB, the product of each segment which is small a multiplied by small b is equal to the product of the segments of the other chord CD which is small c multiplied by small d, okay? So having that in mind, let's go ahead and solve this. So that simply means that 4 multiplied by 36 must be equal to 6 multiplied by this unknown. Let's call it x, alright? Now let's solve 4 times Okay, since this is 6, we can easily cross it out, yes. So to find x, let's divide both sides by 6, right? So divided by 6, divided by 6. Now 6, we remove itself here, we have x, and 6 we take away, 6 we go into 36 6 times. Now 6 times 4 is 24. So it means that the unknown length is 24. If this is the first time you are coming across this lovely channel, please hit the subscription button, turn on your notification bell so that you don't miss our videos. We upload our videos as often as possible. Tune in at your convenience. There must be something new for you to watch. Give us a thumbs up if you love what we are doing. If you are my subscriber, you already know that I love comments. So drop your comments, whatever it is you want me to know. I will definitely be in the comments reading, okay? Thank you very much. So now that we have found the value of our x, let's go ahead and find the radius, okay? Now let's take this to be the center of this circle. If this is the center of the circle, let's drop a perpendicular from the center of the circle to chord BC, okay? Dropping a perpendicular from the center of the circle to chord BC. So we have this. Now remember that a perpendicular to a chord bisects the chord, right? Perpendicular from the center of the circle to a chord bisects the chord. So what that means is that BC is equal to 36 plus four. So it means that BC is equal to 40. Now what this means is that B to this point, let's call this point B, okay? So BP is equal to 20 and PC is also equal to 20, all right? Now if that is the case, if BP is 20, 20 minus four will give us 16. So it means that OP is 16. So from here to here is 16, all right? So we are going to drop another perpendicular from the center of this circle to... We are going to drop another straight line that is coming from the center and perpendicular to A. Okay, let's call this AQ, okay? So if we do that, we are going to have this. So this is also perpendicular to AQ, meaning that it also bisects line AQ. Now let's check. Now line AQ will be six plus 24, meaning that this is equal to 30. Then it implies that A, let's call the meeting point M, okay? So it means that AM, so AM must be equal to MQ, which should be half of 30, which is 15, right? Yes, and if that is the case, it simply means that MQ is 15. All right, so let me remove this 24. We already know this is 24. Let me write it somewhere that OQ is 24 because I know we still need it. Oh, we are enjoying this. Okay, now we have... Remember that from O to P is 16. So definitely from M to the center here should also be 16, right? Since this is right angle, so this is right angle, this is right angle, right? So it means this is a rectangle. So the opposite sides of a rectangle are equal. So here is 16 as well. Now let's join this to Q. Let's join that to Q. Having done that, the center of this set is supposed to be O, but I've named here O, so permit me to name here N. So this is N, all right? Now you can see that MNQ is a right angle triangle. So using Pythagorean theorem, using Pythagorean theorem, we have that MQ squared plus MN squared must be equal to QN squared, all right? Now MQ is 15, MN is 16, is QN squared. So 15 squared is 5 squared. We want to square a number that ends with 5. 5 squared is 25. Now after 1, you'll have 2. 2 times 1 is 2. So it's 2 to 5. I don't like cramming. Yes, I don't like cramming or memorizing. I'd rather have my patterns, which is why I've written a book on mental math. It's basically for multiplication that will give you patterns to multiply anything in your head. I'll find a way to actually bring the book here on YouTube. So look forward to the link that you can use to buy the book from here. So looking at this, we have 2 to 5 plus now 16 squared. We are going to make use of normal mental math. So 6 squared is 36. 1 squared is 1. Write it as a two-digit number. Now 1 times 6 is 6. 6 times 2 is 12. Write it here. You add. So you have 256. So it means that 16 squared is 256. That's another pattern. So we'll write 256 is equal to qn squared. So we add 5 plus 6 is 11. Carry 1. 2 plus 5 is 6. 7 plus 1 is 8. 2 plus 2 is 4. So we have 481 equals qn squared. Now to get qn, we take square root of both sides. And we are going to use the positive square root. That is because we are talking about length and qn cannot be negative. So qn is equal to square root of 481, which is equal to 21.9 units. And this is our radius. So it means that our radius is 21.9. Now the second and most interesting method is actually faster and easier. We are just going to make use of a particular formula. Now looking at this now, this is 16. We don't need this. All we needed is the fact that this length from O to Q is 24. So I'm going to clean everything and leave just the one we need. So from here to here is 24. Now remember that when two chords intersect at right angle in a circle, when two chords intersect at right angle in a circle, then 4 multiplied by the radius of the circle squared is equal to the sum of the squares of all the segments of the chords, right? So if this is, this should be 4r squared is equal to, okay let me explain more. If this is a circle and these two chords intersect at right angle, and this is a, b, c, and d, it simply means that 4r squared will be equal to plus b squared plus c squared plus d squared, okay? That is what I'm trying to explain. Therefore this will be equal to 4 squared plus 6 squared plus 36 squared plus 24 squared. So 4r squared will be 16 plus 36 plus, now let's solve this. You know I don't, I don't memorize or cram. 6 squared is 36, 3 squared is 9, right? That's two digit number. 3 times 6 is 18, 18 times 2 is 36, right? Then we add. We bring down 6, bring down 9, bring down 12. So this is 1296. So using the same formula for 24 squared, 4 squared is 16, 2 squared is 4, right? That's two digit number. 2 times 4 is 8, 8 times 2 is 16. You add 5, 7, 6. So we have 5, 7, 6 as 24 squared. Please give us a thumbs up if you watch to this place, to this level because it simply means that you're enjoying this. So give us a thumbs up, it's free. All right, now we have 4r squared is equal to, let's add. So we have 1296, 5, 7, 6, 36 and 16. 6 into 4 is 24, carry 2. 9 plus 7 is 16, 16 plus 3, 19 plus this is 20 plus 2 is 22. Carry 2 again. 2 plus 5 is 7 plus 2 is 9. Bring down 1. So we have 1, 9, 2, 4. So we are going to divide both sides by 4. So to have r squared, right? So r squared is 4, we go into 19, 4 times, right? That is 4 times 4 is 16, remainder 3. So you go into 32, 8 times, go into 4 once. So you still see that we got r squared is equal to 481. So if you still get a solve for your r, you solve for square root of 481, which gave us I think 21.9 or something in our first method. Thanks so much for watching and see you in my next video. Bye.