Grades 11-12 Video Solutions 2022-2022_11-12

Video Transcription

A rectangle is divided into 11 smaller rectangles, as shown in the diagram. All 11 rectangles are similar to the original large rectangle.  The orientation of the smallest rectangles is the same as the largest. The length of the base of the smallest rectangle is 1. What is the perimeter of the large rectangle?  So here's the diagram a little bit bigger. Now let's give some names to some quantities that we don't know yet. So let's call this side length x and this side length y.  Now all of these rectangles are congruent, and we have 9 of them, so that makes this side length equal to 9x, and we know that this side length is the same as y.  So now we can say this. And since all these rectangles are similar, the ratio of the short side to the long side has to be constant across all of them.  So that means x divided by 1 has to be equal to y divided by 9x, which has to be equal to 9x divided by 2y plus 1, like this.  Now if we look at the first and the last equation, there's an x in both numerators, and we know  that it's not 0, so we can cancel it to get that 1 is equal to 9 over 2y plus 1, which gives us that y equals 4. And now we can substitute back in to calculate x.  If we substitute it in here and multiply by x, we get that x squared equals 4 over 9, so x equals 2 thirds.  Now we just want the perimeter of the large rectangle, which is 9x plus 2y plus 1, and then times 2 to get the other sides.  And so we put in 2 thirds here, 9 times 2 thirds is 6, 2 times 4 for y is 8, and then plus 1, this entire thing is 15, times 2 gets us 30, so our final answer is 30.

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