Geometry problem #133132

The above picture is made up of loads of little squares like the one shown in the top right in red:

Each is half blue. Therefore the area of the blue region of the entire square is half of its area or $\boxed{50}$

Relevant wiki: Area of Figures

Relevant wiki: Length and Area - Composite Figures

look carefully,

there is 25 blue squares....(easily countable)

and there is also 25 white squares..[16 full squares,.............................16 half squares(in top of every side)=8 full squares,...........and 4 quarter square in corners make a full

square;total=25squares]

now, total area=100

.............total squares=50 and blue squares=25

so, blue squares cover $\frac{1}{2}$ of the area or the area of blue squares is=50

The answer is 50 . Notice that the large square has sides of length 10, which is the same as the length of 5 times the diagonal length of the blue square, based on the figure. So, one can directly solve for the side length of the blue square. $10=5l\sqrt{2}\Rightarrow l=2/\sqrt{2}=\sqrt{2}$ . This gives an area of $2$ for a single blue square. Since there are 25 total squares the total area is $25\times 2=\boxed{50}$ .

If the area of the big square is 100, then it's height is 10. Looking at the figure, we see that 5 times the length of the diagonal of a blue square equals 10. Therefore, the length of the diagonal is 2. From this, we deduce that the length of the side of a blue square is $\sqrt{2}$ . Therefore, the total area of the blue squares is $25 \times \sqrt{2} \times \sqrt{2}$ = 50

The area covered by white squares= The area covered by blue squares So area covered by blue squares=100/2=50 sq. units

Since the side lengths are 10, and there are 5 squares connected at the vertices for each side, the diagonal of each blue square is 2. From that info, the side lengths of the blue squares are 2^1/2, and the area is 2. 2×25=50

A rather lengthy solution would be this one:

We know that the side of the big triangle is 10, for 10^2 = 100. Let's call 1/10 of this side "x" = 1. Two "x" form a right triangle with the side of a blue square "y" (see the top right corner).

Since a^2 + b^2 = c^2, that means that x^2 + x^2 = y^2 = 2, the squared side of a blue square. Squaring the side of a blue square naturally gives its area, in this case it must be 2.

Now the total area is 100, there are 25 blue squares, so their area covers half the area of the big square, which is 50.

On a side note: this means that the side of a blue triangle is actually the square root of 2, an irrational number.

I agree with geoff philling. I solved it the same way. The four small triangles in four corners add up to form 1 square.Also the medium sized triangles on each side of the bigger square are taken in pairs to form 1 square each.Also since each blue square is equal in area so the small white square surrounded by 4 blue squares is equal to the area of 1 blue square. Using these rules we get that there are total 50 equal squares forming the big square. This means area of 50 squares = 100. Area of 1 square= 2. No. Of blue squares= 25 Area of blue squares= 25 × 2= 50.

I feel like a complete moron for using trigonometry to solve this. I focused on finding the area of one blue square instead of the bigger picture. Oh well, it was fun ;3

The number of white squares is equal to the number of blue squares across rows and columns = 5. Consequently, blue squares would occupy half the area

There are 50squares each of two unit area Extending this logic there are half blue squares there fore 100/2=50

count the blue diamond and multiply by 2

Shift all the blue square in the white region...u got the answer

I twisted the blue boxes in my mind, and a checker board was left. So, I went with my mathematical gut.