square in square

[khansaheb]

square in square


In the larger square, there is a smaller square formed by lines connecting each corner to the midpoint of one of the sides.

What fraction of the larger square is the smaller filled square? (problem suggested by Rishi Khan)

 

[nasi112]

 

[The Highlander]

Very colourful but what does it tell us???

(I suspect your colouring-in exercise may have slightly adjusted the proportions of the image so it is not immediately obvious to the eye what I think you are trying to hint at. For example, if I transpose the blue triangle over to join the blue trapezium it looks to me that doing so forms an oblong, ie: a non-square rectangle.)

 

[blamocur]

Very colourful but what does it tell us???

(I suspect your colouring-in exercise may have slightly adjusted the proportions of the image so it is not immediately obvious to the eye what I think you are trying to hint at. For example, if I transpose the blue triangle over to join the blue trapezium it looks to me that doing so forms an oblong, ie: a non-square rectangle.)

I am guessing it illustrates that the all areas of the same color are equal.

 

[The Highlander]

I am guessing it illustrates that the all areas of the same color are equal.

Indeed, it does appear to suggest that but what needs to be proved is that they form squares which proof then easily leads to the answer to the original question.

 

[Dr.Peterson]

Very colourful but what does it tell us???

Indeed, it does appear to suggest that but what needs to be proved is that they form squares which proof then easily leads to the answer to the original question.

The colored version can be considered a proof without words. You are expected to think about the picture and see the answer it implies.

I would have given a similar hint (which is all it's meant to be), by suggesting hinges at the midpoint of each side, around which each triangle could be rotated.

In either case, the proof is not hard, and it's even easier just to convince yourself that you could prove it if asked (since no proof was demanded). I see similar triangles all over the place that will do the job.

In fact, it's enough just to visualize extending the picture to an infinite grid of slanted squares and realize that connecting four points as shown would result in exactly what we have (a square with bisected sides).

 

[nasi112]

Very colourful but what does it tell us???

(I suspect your colouring-in exercise may have slightly adjusted the proportions of the image so it is not immediately obvious to the eye what I think you are trying to hint at. For example, if I transpose the blue triangle over to join the blue trapezium it looks to me that doing so forms an oblong, ie: a non-square rectangle.)

It tells you everything.

 

[khansaheb]

square in square

View attachment 39424
In the larger square, there is a smaller square formed by lines connecting each corner to the midpoint of one of the sides.

What fraction of the larger square is the smaller filled square? (problem suggested by Rishi Khan)

1 rectangle + 1 triangle = 1 smaller square
So
5 smaller square = 1 large square
Thus:
(smaller square) : (larger square) = 1:5