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

本站添加: 2026-05-19

[00:00:00]This is indeed a very tricky and confusing math question. Welcome back to my channel. This is not drawn to spell please. And we are told that from here to here is four. From here to here is six. And we 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.

[00:00:30]Um the point of contact, let's call it M.

[00:00:35]All right. Now, first things first, let's extend this line, line A, M. Let's extend it to meet at a point is very important because that is going to help us a lot.

[00:00:58]Okay. So, I don't know if you can see this point, but I think you should. Now, let's call this point point B. All right. This point is point P. Now, if you look at this, let's call this ABC C D E point E. Right? You will see that EP should be equal to MP.

[00:01:21]Yes, because both of them are tangent.

[00:01:26]Both of them are tangent to this circle and they are they meet at a point. Okay.

[00:01:32]When two tangents coming from a circle meets at a point, their length from that point of their from their meeting point are equal. All right? So it means that EP is equal to MP. Now this implies that M is an isosles triangle.

[00:02:03]Isocalles triangle.

[00:02:06]And remember that for an isosles triangle the two the two base angles are equal. For an isocles triangle the two base angles are equal. So this is our theta. That means this place must be equal to theta as well. Now remember that when you have two lines crossing each other. All right? Vertically opposite angles are equal. So if this is theta, this should also be theta. Right? If this is 2 theta, this should also be 2 theta. So if you look at point M you're going to see that the angle here is vertically opposite to this one. So this is also theta.

[00:02:51]Now moving forward moving forward you will notice that this angle here let's call here h.

[00:03:08]So angle at point h is also theta. I will explain this is also theta. I will explain. Now if you look at a hash, okay, a hash and you look at h e h e. And let's draw e.

[00:03:27]E. All right. This is e.

[00:03:33]This is h e. And this is a. All right.

[00:03:38]Now this we are told that this is a square meaning that a h is what parallel to e 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 is a transversal and we have here as theta then definitely here should also be theta at H 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're 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're going to see me so you need to subscribe so like you see me all the time and always comment on our videos.

[00:04:29]We must reply and we are there at the comment section. All right, so my shout out today goes to Mustafa 9052.

[00:04:38]Thank you for always stopping by and for your lovely comments. Now um looking at this now let's go ahead and establish some facts and find the area of this square.

[00:04:57]Now if we say that EP is equal to MP all right 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?

[00:05:18]Angle angle similarity using angle angle similarity similarity similarity whatever. So using angle angle similarity um you're going to notice that triangle A HM and triangle P E M or M E P are similar.

[00:05:57]They are similar.

[00:05:59]And remember that similar triangles are what? The ratio of the corresponding sides of similar triangles are equal.

[00:06:09]Ratio of corresponding sides of similar triangles are equal. Now it simply means that if we say EP over a H# is equal to E M over M H or H which is equal to 6 / 4 which is equal to 3 / 2. All right. So it means that the side is what in the ratio of 3 is to 2.

[00:06:48]Yeah. So if we call ep 3 y, it simply means that a h is what? 2 y.

[00:07:02]So let's call this 3 y. Then a h is 2 y.

[00:07:08]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 hm triangle a hm is an isosles triangle.

[00:07:27]An isosles triangle.

[00:07:33]Yes. Since the two base angles are equal it is an isocles 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.

[00:07:49]So we have 2 y here as well. So this side is equal to this side.

[00:07:57]Now using this particular theorem that we discussed in the beginning. All right.

[00:08:04]You can see that AM and A, let's call this point of contact F. I think do we have F here? No, let's call it F. Okay, a M should be equal to AF. Why? Because both of them are tangent coming from this meeting at a point A tangent to this circle meeting at a point A. So AF should be equal to A M. Right?

[00:08:32]So this place is also 2 y.

[00:08:36]Now let us form a square. Let us form a square here. Let's join this to the center of this circle.

[00:09:00]Okay. So it simply means that from here to here is the radius of this circle.

[00:09:08]From here to here this is oh is the radius of this circle. This is radius and this is radius.

[00:09:14]All right we're almost there.

[00:09:22]considering triangle A B.

[00:09:26]So let's draw that out.

[00:09:30]I want to minimize my space.

[00:09:34]So this is A, this is B and this is P.

[00:09:38]A B P. All right. So you can see that BP is R + 3 Y.

[00:09:48]A B is 2 Y + I.

[00:09:52]This is 90°. And A is 2 Y. A is 2 Y + 3 Y, which is 5 Y.

[00:10:01]Now using Pythagorean theorem we are going to have 2 y + r all 2 + r + 3 y all squared will give you 5 y 2.

[00:10:21]So recall that a + b all squ is a 2 + 2 a b + b 2.

[00:10:32]So we are going to write this as 2 y 2 that is 4 y 2 + 2 * 2 y which is 4 y * r. So it's going to be 4 y r + r 2 + then this will give us r 2 + 2 * 3 * 2 * r * 3 y will give us 6 r y + 3 y 2 is 9 y 2 is equal to 5 y^2 will give us 25 y^ 2.

[00:11:16]Okay, I'm trying to minimize my space.

[00:11:19]So we have 4 y^ 2 + 9 y^2 that is 13 y 2 + 4 y r + 6 y 6 r y or 6 y r is 10 y r + r 2 + r 2 is 2 r 2 is = 25 y^ 2.

[00:11:51]If we transfer this 13 y^ 2 to this side, we are going to have 10 y r + 2 r 2 is equal to 25 y^2 - 13 y^ 2, which is 12 y^ 2. Now let's divide through by 2.

[00:12:12]/ 2 / 2 / two and we will have.

[00:12:30]So this will be 10 y r this will be r 2 sorry this will be 5 y r + r 2 is = 6 y 2. Now let's transfer everything to this side. If we do that we have 6 y^ 2 - 5 y r - r 2 is 0. Now let's replace - 5 y with - 6 y r - y r. Right? So we can actually write this as 6 y^ 2 - 6 y r - y r then - r 2 is 0. So let's factoriize 6 y is common here. We bring it out.

[00:13:21]Then we have y - r remaining right minus what is common here is r. We open our our brackets. Sorry this is going to be - 6 y r + y r rather because we have - 5 r here. So this is positive. Now when y r is divided + r we left with y - r² / r is - r is z. So from here we have this in common right? So factoring that out we going to have 6 y + r * y - r is = 0.

[00:14:06]So this implies that 6 y + r is = 0 and that y - r is = 0.

[00:14:20]So moving forward from the left we will see that 6 y is equal to r and this is not possible because our r is radius it can't be negative.

[00:14:32]So we ignoring this and using this that we will get from here that y is equal to r.

[00:14:42]Anywhere I see y now I'm replacing it with r. Okay. So this 2 Y becomes 2 R.

[00:14:48]This 2 Y here becomes 2 R.

[00:15:01]All right. Now let me join center O to this point perpendicular line.

[00:15:13]So we have angle 90°.

[00:15:17]Now let's consider triangle.

[00:15:19]Let's call the 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.

[00:15:36]We have another triangle K H.

[00:15:46]and K is going to give us remember that from A to H is 2 R. All right. And A K is what? R. So it means that K H is 2 R - R which is R. Right? So here is R.

[00:16:03]This length from here to here is same thing as 2 R + R. So that is 2 R + R which is 3 R.

[00:16:13]Good. Now from here to here which is H E is 10 and this is 90°. So using the Pythagorean theorem using the Pythagorean theorem you will see that 3 R² + R² is equal to 10². So we have 9 r² + r² = 100 10 r² = 100ide by 10 / 10 and you have that r² is= 10. So we keeping that r² is= 10.

[00:17:01]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 square and one length of this is 2 arrow + r right which is 3 arrow so it's going to be 3² which is 9 arrow which is 9. Pardon me it's raining but I need to finish this video. Yes. So r square is already 10.

[00:17:31]So it means that the area of this square is 90 unit squared.

[00:17:40]Thank you so much for watching and see you in my next video. Bye.