Brainsteem Mathematics Challenges: Squares in Triangles
Today, an amusing but fairly straight-forward geometry puzzle.
Below is a large square with a diagonal contructed from corner to corner. Within each of the triangles thus created, we have constructed two smaller squares, A and B, in two different orientations.
The question is to find the ratio, (area A) : (area B)
The first correct answer and further interesting comments will be rewarded with an upvote.
Enjoy!
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@rycharde manages the AAKOM project and the MAP forum.
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Area of A is a quarter of the main square, which is four and half times of the area of B. So the ratio of area A : area B is 1/4 : 1/4.5 or 9:8
:-) how do we know the area of B? (just to help those for whom it is not "obvious"!)
Inside the largest right-angled triangle (half the size of the main square) where B is located, the two right-angled triangles adjacent to B are equivalent to cutting B diagonally. So their combined area is the same as B. The smallest triangle at the top of B is equivalent to a right-angled triangle made by cutting B along two diagonals crossing with each other. Therefore, there are 1 + 1 + 1/4 = 2.25 units of B in the "half-the-size-of-main-square" right-angled triangle. In other words, there are 2.25 x 2 = 4.5 units of B in the main square.
Sorry for the clumsy explanation. It should be much clearer in diagrams.
The fun of this puzzle is to find the trick of solving it without too many calculations.
Yes, in the UKMT Maths challenges there is always at least one geometry question that is much faster solved by "moving areas" than calculations.
In this case, as you said, chop up the two half-squares into triangles and it becomes clear!
The ratios then become: 2/4 : 4/9, or 9:8.
Don't worry about the LE, it's something I do automatically since I'm from Germany. It means Längeneinheiten which means length units.
That's OK, thanks - luckily, just got in time to give you a small upvote :-)
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Area of A will be a fourth of the fundamental square, which is four and half circumstances of the area of B. So the proportion of territory A : area B is 1/4 : 1/4.5 or 9:8
Wow! Apart from changing the way you write the numbers, your answer approach and sentence structure are exactly the same as mine. If you are a true maths lover perhaps you can consider to do your own work in future.
Unfortunately, I've gotten here a bit late. I only wish I could have discovered this page earlier.
Others have already solved this problem in their own ways, so I'll let their solutions stand. Perhaps next time...