This Geometry Puzzle Will Play On Your Mind Till You Think You Are Crazy

Added: 2026-05-09

[00:00:00]In this video, we are required to find the length and width of the outer rectangle. So, we have a rectangle and within the rectangle, there are different rectangles and one square. So, this rectangle has an area of 14 while this one has an area of 15. This one has an area of 17. This rectangle has an area of 12. And this middle part is where we have the square which has an area of 1.

[00:00:30]So with that what is the area sorry length and width of this outer rectangle.

[00:00:38]So first let's call this rectangle label the points let this be a b c and this d.

[00:00:48]Now first if we are to look at the square.

[00:00:55]So area of square we know that area of a square is given by side squared but our area was given as 1. So 1 is equ= to side squared. So side square is equ= to 1. We find the square root on both sides. So s square root of 1 is 1. So each side of the square is one. Okay. So that implies that from here to here is one. Also from here to here is one. Also from here to here is one. Also from here to here is one.

[00:01:33]Now let's label all our rectangles that are inscribed for easier identification.

[00:01:39]So let this be one. Let this be two. Let this be three. and let this be rectangle four.

[00:01:54]Now if we are to first first of all focus on rectangle 4 this one here let its length be x. So we are letting from here to here we are letting it to be x. Okay we know that area of a rectangle is equals to length * width and we know the area is 12. So 12 is equals to the length x the width which we do not know.

[00:02:37]Therefore, since we want to find the width, make it the subject will be equals to 12 / x. So, from here to here, we've got it to be 12 over x.

[00:02:55]Next, let's consider rectangle 1.

[00:03:04]Okay.

[00:03:07]Now we know that for rectangle one we know that from here to here is x. Okay from this point to this point is x and from here to here is one implying that this is the width of this rectangle. Okay let's say the length this is the length of this rectangle one. So that implies that from here to here will be x - 1. Okay.

[00:03:43]So area of a rectangle is length * width. We know it's area is 14= to the length. We've got it as x - 1 * the width which we don't know. So w will be = to 14 / x -1.

[00:04:05]So from this point to this point is 14 / x -1.

[00:04:14]Next let us consider rectangle 3.

[00:04:23][snorts] Now in in rectangle 3 since from here from this point to this point is 12 /x and from here to here is 1. So meaning from this point to this point will be 12 12 x + 1. Okay. So here to here is 12 x + 1. But let's simplify it. So let's use crisscross method. So 12 * 1 is 12 + x * 1 is x / x * 1 which is x. So we have 12 + x / x as the width of this rectangle 3.

[00:05:13]12 + x / x.

[00:05:18]We know that area is length time width.

[00:05:23]We already have our area as 17. So 17 is equals to length which we don't know time this width 12 + x / x.

[00:05:36]So we make l the subject. So length will be equals to 17 / 17 over this which is justide 12 + x / x. Therefore, L will be = 17 * the reciprocal of this. So, * X / 12 + X.

[00:06:01]Therefore, length will be = 270 * X is 17 X / 12 + X.

[00:06:10]Therefore, from here to here is 17 x over 12 + x.

[00:06:22]Finally, looking at rectangle 2.

[00:06:33]So in this rectangle two from here to here is one which is the side of the square from here to here. And from here to here we've got it as 17 x / 12 + x.

[00:06:47]That means that the length of this rectangle is 1 + 17 / 12 + x.

[00:06:57]So let us simplify this. This will be equals to.

[00:07:01]So if this is over one crissross 1 * this shall have 12 + x then + 17 x over 12 + x.

[00:07:21]Okay. So this will be equals to open this bracket 12 + x + 17 x [snorts] over 12 + x which will be equals to 12 + x + 17 x is 18 x over 12 + x.

[00:07:44]Therefore, the length of rectangle 2 from here to here is 12 + 18 x over 12 + x.

[00:07:58]And then for the width of this same rectangle will be so from this point to this point we know it's 14 / x -1 while from this point to this point we know it's 1. So the width of this rectangle will be 14 / x - 1 then minus this one to get just here from here to here. So minus 1.

[00:08:27]So we simplify this crisscross method.

[00:08:32]This will be equals to 14 - 1 into x -1 / x - 1.

[00:08:47]So this will be equals to 14 - x. This times this becomes + 1 / x - 1. So this is equals to 14 + 1 is 15 - x over x - 1. So that's the width. So from here to here is 15 - x over x - 1.

[00:09:22]So now since we have length and width we know that area is equals to length * width and we know area of this one is 15. So 15 is equ= to length which is 12 + 18 x / 12 + x [snorts] time width which is 15 - x over x - 1.

[00:09:56]Now let's multiply by 12 + x into x -1 on both sides. So we multiply this side here time 12 + x into x -1 also * 12 + x into x -1 on this side as well. So on this side this cancels with this and this cancels with this. So we left with 15 into 12 + x into x -1 being equals to just the numerators 12 + 18 x into 15 - x.

[00:10:48]So now we solve for what is in the brackets. So 15 * 12 is 180 + 15 * x is 15 x. Okay. Then * x - 1. This is equ= to 12 + 18 x into 15 - x. Okay. Now 180 * x is 180x.

[00:11:16]180 * -1 is - 180.

[00:11:21]15 x * x is + 15 x^ 2ar 15 x * -1 is - 15 x this is equals to 12 * 15 is 180 12 * -x is - 12 x 18 x * 15 is + 270x then 18 x * -x is - 18 x^² so so we rearrange and bring everything on one side so that we equate the equation to zero. So we shall have this 15 15 x².

[00:12:14]When this 18 x² crosses it becomes + 18 x² then + 180x.

[00:12:25]Okay. - 15x.

[00:12:30]Then when this crosses it becomes + 12 x. When this crosses it becomes - 270x then - 180 then when this crosses it becomes - 180 this is equals to zero.

[00:12:50]Okay.

[00:12:52]So 15 x^2 + 18 x 2 gives us 33 x 2. Then 180x - 15x + 12x this here 180 - 15 + 12 - 270 gives us - 93x then - 180 - 180 gives us - 360. This is equals to zero.

[00:13:24]So we divide everything by three since they're all divisible by three.

[00:13:31]So here we get this by 3 gives us 11 x^2 minus this gives us 31xus this gives us 120. This is equals to 0.

[00:13:46]As you can see we come up with a quadratic equation and we're going to solve it using the quadratic formula.

[00:13:52]So our value of a here is 11, our b is -31 and our c is 120.

[00:14:03]So the quadratic formula is x will be = - b + or minus<unk> b ^ 2 - 4 a c over 2 a.

[00:14:16]So x will be equ= to minus our b is -31 + or minus<unk> of -31^ 2 - 4 * 11 * 120 over 2 * a which is 11.

[00:14:42]So x is = to * negative is positive. So 31 + or minus 31 -31 * -31 we get 9 61.

[00:14:59]Then -4 * 11 gives us -44.

[00:15:05]- 44 * -1 120 gives us positive 528 0. This is over 2 * 11 which is 22.

[00:15:19]So x is equ= to 31 + or minus roo<unk> of this plus this gives us 6 2 4 1 over 22. So x = 31 + or minus sorry x = 31 [snorts] plus or minus square<unk> 6 2 4 1 is 79.

[00:15:51]So this over 22.

[00:15:55]Therefore, x will be equals to either 31 + 79 / 22 or 31 - 79 / 22. In this case, x will be 110 / 22. And in this case, x will be - 48 / 22.

[00:16:19]So, x for here when you divide this, you get five. And when you divide this you get 2.182.

[00:16:29]Now since x represents distance our value of x cannot be negative because distance cannot be negative. Therefore we reject this solution and consider x is equals to five as our value of x.

[00:16:48]Now that we have got the value of x, next thing is just replace for x in this what we have to find the length and width of a b c d. So first let's start with length.

[00:17:03]So the length of a b c d rectangle ab cd.

[00:17:11]So length will be equals to BC which is X X + 17X over 12 + X and we've got our X as 5. So just replace this will be equals to 5 + 17 * 5 / 12 + 5.

[00:17:40]So this is equals to 5 + 17 * 5 is 85 over 12 + 5 is 17. 17 goes into 85 5 times. So this will be equals to 5 + 5.

[00:17:55]So our length is 10.

[00:17:58]Okay. Length of ABC D is 10 units.

[00:18:02][clears throat] Next, let's find the width of rectangle A B C D.

[00:18:11]Now, the width will just be this DC, which is this plus this.

[00:18:20]So our width will be 15 - x / x - 1 + 12 + x / x.

[00:18:34]Okay, since we already know our value of x is 5, this will be 15 - 5 / 5 - 1 + 12 + 5 / 5, which is equals to 15 - 5 is 10. So 10 / 5 - 1 is 4 + 12 + 5 is 17 over 5.

[00:19:01]So we use crissross method.

[00:19:05]So this will be equals to 10 * 5 + 4 * 17 over 4 * 5 which is equals to 10 * 5 is 50 + 4 * 17 is 68 over 4 * 5 is 20.

[00:19:34]So this is equals to 50 + 68 gives us 118 over 20 which is approximately equals to 5 9.

[00:19:51]So this implies that our length.

[00:19:55]So for rectangle A B C D our length is equals to 10 units and our width is equals to 5.9 units. And this is what they wanted us to find. Yeah. Thank you so much for watching. Hope you enjoyed this video.

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