A tricky Geometry math question|solve for x.

Hi, let's find the value of x. Welcome back to my channel. So in this video we are asked, this is a semicircle but this is not drawn to scale please. So in this semicircle we are told that the length a b is 24 and length b c is 7. We are asked to find length c d and this is a quarter circle sorry not a semicircle a quarter circle and this d is what the center of the quarter circle. Now first of all let's join a to c. If we do that we form triangle a b c which is a right angled triangle. Therefore using the Pythagorean theorem in triangle a b c using Pythagorean theorem using Pythagorean theorem we have that a b squared plus b c squared is equal to the hypotenuse squared which is a c squared. Okay so a b is 24 so we have 24 squared plus b c 7 7 squared is equal to a c squared. Now 24 squared is I don't memorize that but I'm going to use a math trick to find it. 4 squared is 16. 2 squared is 4. Alright that's a two digit number 0 4. 2 times 4 is 8 and 8 times 2 is 16. So if I add that we'll have 5 7 6. Alright so that means that 24 squared is 576. So we have 5 7 6 plus 7 squared is 49. Alright is equal to a c squared. Now if we add these together 9 plus 6 is 15. Carry 1. 7 plus 4 is 11 plus 1 12. Carry 1 again we have 6 2 5 is equal to a c squared. If we take square root of both sides so we have square root of 6 2 5 is equal to square root of a c squared. So this we remove this we have a c is equal to now square root of 6 2 5 is 15 uh 25 sorry almost everybody knows that is 25 right. So that means our a c is 25. Alright if this is the first time you have seen this lovely and amazing channel and this lovely face please click the subscribe button turn on your bell so that you don't miss our amazing videos. We upload our videos as often as possible. Tune in at your convenience there must be something new for you to watch. Thank you. Okay so looking at this what if we decide to complete this and make this a semicircle and then complete this particular line. So we now have something like this. You know that this is the center of this circle right. So if we call this e it means that a d is equal to d e because both of them are the radius of the circle right. So this place will be equal to this place. Alright and being that this is the center of this circle this is a right angle right and this is also a right angle right. Now we have two triangles triangle a c d and triangle c d e. Okay now let's compare. Comparing triangle a c d and triangle c d e. Okay now comparing both of them what do we notice. We notice that they have equal length. Length a d is equal to length d e and they share a common angle. Okay they have equal angle which is angle d is common to both of them. This is a d is equal to c d e. They have 90 degrees each. So this simply means that triangle a c d is similar to triangle c d e. So if they are similar and this side is equal to this side they share the same triangle it simply means that c e is also 25 right. Okay now moving forward remember that our intention is to find the value of x. Now moving forward we have triangle a b e right. In triangle a b e it is a right angle triangle using the Pythagorean theorem we have a b squared plus b e squared should be equal to a e squared. Now we have 24 squared plus b e is what 7 plus 25 and that will give us 32 right. 32 squared is equal to a e squared. Now 24 squared we got that as what did we get it as 500 and what sorry. So let's do that 24 squared 4 squared 16 2 squared 0 4 2 times 4 8 8 times 2 16. Okay we got 5 7 6. So we have 5 7 6 plus now 32 squared 32 squared using the same math trick 2 squared is 4 we write it as two digit number. Anytime you are using this trick ensure that any number you are getting you are writing it as a two digit number. Okay so 2 squared is 0 4 3 squared is 9 which is 0 9 3 times 2 is 6 6 times 2 is 12. So we add we have 4 we have 2 we have 10. So what we have here is 100 and 1024. So we have 1024 is equal to a e squared. Now in essence let's add 5 7 6 to 1024. Now 76 and 24 will give us 100 right. So if we add the 100 to 5 that means to 500 we have 600. That means this plus this will give us 1600 is equal to a e squared. Now if we take square root of both sides so we have square root of 1600 is equal to square root of a e squared. Now this remove this we have a e is equal to square root of 1600 is 40. Okay so we can see that a e is equal to 40. Okay what are we doing with it you are going to see pretty soon. But a e is equal to the diameter of this circle right. So it means that half of a e which is half times 40 will give us 20 right which is the radius of this circle. So what this means is that d is 20 and 8 is 20. So here we have 20 and we have 20. All right amazing. Okay if we continue we are there almost. To get the value of x is now easy. Let us take triangle a c d or triangle c d e. So if I draw that out c d is the x right d e is 20 and c e is 25. And this is a right angle. Now using Pythagorean theorem we have 25 squared is equal to 20 squared plus x squared. So this we come here we have 25 squared minus 20 squared is equal to x squared. So now we have 25 minus 20 multiplied by 25 plus 20 using a difference of two squares is equal to x squared. So 25 minus 20 is 4. 4 and 25 plus 20 will give us 45. We have 45 is equal to x squared. And now if we multiply further 4 times 5 is 20. So rise 0 and carry 2. 4 times 4 is 16 plus 2 is 4 times 5 is 20. Oh see we are making the mistake. 25 minus 20 is 5. Okay so 5 times 5 is 25. Carry 2. 5 times 4 is 20 plus 2 is 22 is equal to x squared. So if we take square root of both sides this we remove this and we have 15 is equal to x. So it means that x is equal to 15. Alright thank you for watching and I wish and would love to see you in my next video.