This Geometry Puzzle Looks Hard… Until You See the Trick!

This is indeed a very tricky and confusing math question. Welcome back to my channel. This is now drawn to scale, please. And we are told that from here to here is 4, from here to here is 6. And we are expected to find the area of this square. Please, this is meant to be a square. So, we are asked to find the area of this square. Let's name the points A, B, C, D. The point of contact, let's call it M. Now, first things first, let's extend this line, line A, M. Let's extend it to meet at a point. It's very important because that is going to help us a lot. So, I don't know if you can see this point, but I think you should. Now, let's call this point, point B. This point is point B. Now, if you look at this, let's call this A, B, C, D, E, point E. You will see that EP should be equal to MP. Yes, because both of them are tangents. Both of them are tangents to this circle and they meet at a point. When two tangents coming from a circle meet at a point, their lengths from their meeting point are equal. So, it means that EP is equal to MP. Now, this implies that MEP is an isosceles triangle. And remember that for an isosceles triangle, the two base angles are equal. For an isosceles triangle, the two base angles are equal. So, this is theta. That means this place must be equal to theta as well. Now, remember that when you have two lines crossing each other, vertically opposite angles are equal. So, if this is theta, this should also be theta. If this is 2 theta, this should also be 2 theta. So, if you look at point M, you are going to see that the angle here is vertically opposite to this one. So, this is also theta. Now, moving forward, you will notice that this angle here, let's call here H. So, angle at point H is also theta. I will explain. This is also theta. I will explain. Now, if you look at A-H and you look at H-E, and let's draw EP. This is EP, this is H-E, and this is A. Now, we are told that this is a square, meaning that A-H is what? Parallel to EP. And if that is the case, this simply means that H-E is a transversal. Yes, it means that H-E is a transversal. Now, if H-E is a transversal and we have K as theta, then definitely here should also be theta at H-E because these two angles are alternate angles, right? And alternate angles are equal. So, that is why this place is theta as well. If this is the first time you are coming across my lovely channel, please click the subscription button and turn on your notification bell. Yes, because if you don't subscribe, this might be the last time you are going to see me. So, you need to subscribe so that you will see me all the time. And always comment on our videos. We must reply and we are there at the comment section. Alright, so my shout out today goes to MustafaDal9052. Thank you for always stopping by and for your lovely comments. Now, looking at this now, let's go ahead and establish some facts and find the area of this square. Now, if we say that EP is equal to MP, alright? And that these two angles are equal to these two angles. What does that mean? It simply means that if these two angles are equal, using angle-angle similarity, right? Angle-angle similarity. Using angle-angle similarity. Similarity. Similarity. Whatever. So, using angle-angle similarity, you are going to notice that triangle A-H-M and triangle P-E-M or M-E-P are similar. They are similar. And remember that similar triangles are what? The ratio of the corresponding sides of similar triangles are equal. Ratio of corresponding sides of similar triangles are equal. Now, it simply means that if we say EP over A-H-M is equal to E-M over M-H-M, which is equal to 6 over 4, which is equal to 3 over 2, alright? So, it means that the side is what? In the ratio of 3 is to 2. Yeah. So, if we call EP 3-Y, it simply means that A-H is what? 2-Y. So, let's call this 3-Y. Then A-H is 2-Y. Now, remember that MP is equal to EP. So, this side is also 3-Y, right? Now, I want you to note that triangle A-H-M is an isosceles triangle. An isosceles triangle. Yes. Since the two base angles are equal, it is an isosceles triangle, meaning that the two sides are equal as well. So, A-M should be equal to A-H, which is equal to 2-Y. So, we have 2-Y here as well. So, this side is equal to this side. Now, using this particular theorem that we discussed in the beginning, alright? You can see that A-M and A, let's call this point of contact F. Do we have F here? No. Let's call it F, okay? A-M should be equal to A-F. Why? Because both of them are tangents coming from this, meeting at a point A, tangents to this circle, meeting at a point A. So, A-F should be equal to A-M, right? So, this place is also 2-Y. Now, let us form a square. Let us form a square here. Let's join this to the center of this circle. Okay. So, it simply means that from here to here is the radius of this circle. From here to here, this is O, is the radius of this circle. This is radius and this is radius, alright? We're almost there. Considering triangle A-B-P. So, let's draw that out. I want to minimize my space. So, this is A, this is B, and this is P. A-B-P, alright? So, you can see that B-P is R plus 3-Y. A-B is 2-Y plus R. This is 90 degrees. And A-P is 2-Y. A-P is 2-Y plus 3-Y, which is 5-Y. Now, using Pythagorean theorem, we are going to have 2-Y plus R, all squared, plus R plus 3-Y, all squared, will give you 5-Y squared. So, recall that A plus B, all squared is A squared, plus 2-AB plus B squared. So, we are going to write this as 2-Y squared, that is 4-Y squared, plus 2 times 2-Y, which is 4-Y, times R. So, it's going to be 4-Y R, plus R squared, plus, then this will give us R squared, plus 2 times R times 3-Y will give us 6-R-Y, plus 3-Y squared is 9-Y squared, is equal to 5-Y squared will give us 25-Y squared. I'm trying to minimize my space. So, we have 4-Y squared, plus 9-Y squared, that is 13-Y squared, plus 4-Y R, plus 6-R-Y, or 6-Y R, is 10-Y R, plus R squared, plus R squared, is 2-R squared, is equal to 25-Y squared. If we transfer this 13-Y squared to this side, we are going to have 10-Y R, plus 2-R squared, is equal to 25-Y squared minus 13-Y squared, which is 12-Y squared. Now, let's divide through by 2, divided by 2, divided by 2, divided by 2, and we will have, so this will be 10-Y R, this will be R squared, sorry, this will be 5-Y R, plus R squared, is equal to 6-Y squared. Now, let's transfer everything to this side. If we do that, we have 6-Y squared, minus 5-Y R, minus R squared, is 0. Now, let's replace minus 5-Y with minus 6-Y R minus R-Y R, right? So, we can actually write this as 6-Y squared, minus 6-Y R, minus Y R, then minus R squared is 0. So, let's factorize, 6-Y is common here, we bring it out, then we have Y minus R remaining, right? Minus, what is common here is R, we open our brackets, sorry, this is going to be minus 6-Y R plus Y R, rather, because we have minus 5 R here, so this is positive. Now, when Y R is divided by plus R, we are left with Y, minus R squared divided by R is minus R, is 0. So, from here, we have this in common, right? So, factoring that out, we are going to have 6-Y plus R, multiplied by Y minus R, is equal to 0. So, this implies that 6-Y plus R is equal to 0, and that Y minus R is equal to 0. So, moving forward, from the left, we will see that 6-Y is equal to negative R, and this is not possible because R is radius, it can't be negative. So, we are ignoring this, and using this, that we will get from here that Y is equal to R. Anyway, I see why now I'm replacing it with R, okay? So, this 2-Y becomes 2-R, this 2-Y here becomes 2-R. Alright, now, let me join center O to this point, perpendicular line. So, we have angle 90 degrees. Now, let's consider triangle, let's call their point of contact K. Do we have K here? No, let's call their point of contact K. So, we are going to have another triangle, we have another triangle K-H-E, and K-H is going to give us, remember that from A to H is 2-R, alright? And, A-K is what? R. So, it means that K-H is 2-R minus R, which is R, right? So, here is R. This length from here to here is the same thing as 2-R plus R. So, that is 2-R plus R, which is 3-R. Good. Now, from here to here, which is H-E, is 10, and this is 90 degrees. So, using the Pythagorean theorem, using the Pythagorean theorem, we will see that 3 R-squared plus R-squared is equal to 10-squared. So, we have 9 R-squared plus R-squared is equal to 100, 10 R-squared is equal to 100, divide by 10, divide by 10, and you have that R-squared is equal to 10. So, we are keeping that the R-squared is equal to 10. So, remember that we are asked to find the area of this square, and as a result of that, because this is a square, area of a square is length squared, and one length of this is 2-R plus R, right? Which is 3-R. So, which is 9, pardon me, it's raining, but I need to finish this video. Yes. So, R-squared is already 10. So, it means that the area of this square is 90 units squared. Thank you so much for watching and see you in the next video. Bye.