Can you solve this?

[00:00:00]Here's an interesting geometry question for you. We are given a large blue rectangle made up of several squares of different sizes. The only information provided is that the small green square at the center has an area of nine square units. Using just this information, our task is to find the area of the entire blue rectangle.

[00:00:22]Remember that the figure is not necessarily drawn to scale. So, we must rely only on the given information.

[00:00:30]To find the area of the blue rectangle, we first need to know its length and width because the area of a rectangle is equal to length multiplied by width.

[00:00:40]The green square has an area of nine square units. We know that the area of a square is equal to its side length squared. Therefore, the side length is equal to the positive square root of nine, which is three units. Since every side of a square is equal, we label all four sides of the green square as three units. Now, look at the two squares at the bottom of the figure. Since both of these squares have the same height, therefore they have the same side length and thus they are identical.

[00:01:13]Let us assume that the side length of each square is A because all sides of a square are equal. Every side of these two squares is also A.

[00:01:24]Next, focus on the square located on the bottom left side of the diagram. Its side is made up of two smaller parts.

[00:01:32]One part is the side of the green square, which is three units, while the other part is the side of the smaller square, which is A.

[00:01:42]Therefore, the total side length of this square is A plus three.

[00:01:47]Therefore, we label all four of its sides as A plus three. Now, let us calculate the width of the blue rectangle. The bottom side of the rectangle made up of three consecutive lengths.

[00:02:00]A + 3, A, and one more A.

[00:02:05]Adding these three lengths together gives us a total width of 3A + 3.

[00:02:11]Next, let us move to this square.

[00:02:14]One part of its side is the square, which is A + 3. The remaining part is the side of the green square, which is three units. Therefore, the side length of this square is A + 6. So, we label every side of this square as A + 6. Now, focus on the large square in the upper right corner.

[00:02:36]One part of its side is the side of the previous square, which is A + 6. The remaining part is the side of the green square, which is three units. Therefore, the side length of this square becomes A + 9.

[00:02:52]Once again, because it is a square, all four of its sides are equal to A + 9.

[00:02:59]Let us now find the height of the blue rectangle.

[00:03:03]Looking at the right side, the total height is made up of the upper square with side length A + 9 and the lower square with side length A.

[00:03:13]Adding these together gives a total height of 2A + 9.

[00:03:18]We will now verify this using the left side.

[00:03:21]Looking at the left side of the blue rectangle, the height consists of the upper square with side length A + 6 and the lower square with side length A + 3.

[00:03:33]Adding these two lengths together also gives 2A + 9.

[00:03:39]This confirms that both sides of the rectangle have the same height, exactly as expected.

[00:03:46]Our calculations are therefore consistent.

[00:03:49]Now, let us find another expression for the width of the blue rectangle.

[00:03:54]Looking across the top, the width is formed by the upper left square with side length A + 6 and the upper right square with side length A + 9.

[00:04:05]Adding these two lengths gives 2A + 15.

[00:04:10]Earlier, we found that the width was equal to 3A + 3.

[00:04:15]Since both expressions represent the same width, they must be equal.

[00:04:20]Therefore, 3A + 3 is equal to 2A + 15.

[00:04:25]Subtract 2A from both sides to obtain A + 3 is equal to 15.

[00:04:32]Next, subtract 3 from both sides. This gives us A is equal to 12.

[00:04:39]Now, substitute the value of A into the expressions for the dimensions of the rectangle.

[00:04:45]The width is equal to 3 multiplied by 12 + 3, which gives 39 units.

[00:04:52]The height is equal to 2 multiplied by 12 + 9, which gives 33 units.

[00:04:58]Therefore, we multiply 39 by 33, which is this.

[00:05:04]Hence, the area of the blue rectangle is 1,287 square units.

[00:05:11]This is our final answer.

[00:05:13]That was super duper cool. Like, share, and subscribe. So good.