Area of one of four regions within a rectangle

Question

There is a figure below (a rectangle). You can see different colors depicting different regions of the figure. The labels on the top of a region defines the area of that region.

Can you find the area of the green shaded region labelled with a question mark?

Source: http://gpuzzles.com/mind-teasers/mathematical-picture-area-problem/

Answer 1

This problem can be solved with similar triangles property.

The White triangle is similar to the Red one Now Ratio of sides is $2:3$ (as the ratio of areas = ratio of bases if the height is same)

Seeking that, area of the white triangle $= \frac{9}{4} \times$ area of the red triangle $= \frac{9}{2}$ Following all the above, the Green region = $5.5$ units

Answer 2

This problem can be solved with similar triangles property.

Answer 3

Combine the red and blue triangles to form a red-blue 45-45-90 isosceles right triangle and assume that the two sides equal 3.1622 cm and that the hypotenuse is 4.47203 cm with an area of 5 cm².

Split the 90 degree angle of the combined red-blue isosceles right triangle into two angles of 56.31 degrees and 33.69 degrees to form two triangles with an area ratio of 3 cm² and 2 cm². $$A = 1/2 * bh$$ since A = 3 - 2.50 because 2.50 is the area when the red-blue isosceles right triangle is split into two equal parts
$$0.50 = 1/2 * b * 2.236$$ $$b = 1 / 2.236$$ $$b = 0.4472$$ $$tan(x) = b / h$$ $$tan(x) = 0.4472 / 2.236$$ $$tan(x) = 0.2$$ $$x = arctan(0.2)$$ $$x = 11.31^o$$
blue scalene angle = 45° + 11.31°
blue scalene angle = 56.31°

red scalene angle = 90° - 56.31°
red scalene angle = 33.69°

blue triangle = 56.31° to form an area of 3 cm²
red triangle = 33.69° to form an area of 2 cm²

3 cm² blue triangle:
45° + 56.31° = 101.31°
180° - 101.31° = 78.69°

2 cm² red triangle:
45° + 33.69° = 78.69°
180° - 78.69° = 101.31°

3 cm² blue triangle : 45° - 56.31° - 78.69° degree scalene triangle
2 cm² red triangle : 45° - 33.69° - 101.31° degree scalene triangle

to calculate the area of the white triangle, we combine the blue and the white triangle which becomes a blue-white right triangle with a apex angle of 56.31°. after getting the area of the blue-white triangle, we subtract the blue area which is 3 cm² to get the area of the white triangle.

tan(56.31°) = long side of the white triangle / 3.1622
long side of the white triangle = tan(56.31°) * 3.1622
long side of the white triangle = 4.7433 cm

Calculating the area of the rectangle:
area of the rectangle = 4.7433 * 3.1622 = 15 cm²

area of the combined blue-white right triangle = area of the rectangle / 2 = 15 / 2
area of the combined blue-white right triangle = 7.50 cm²

area of the white triangle = area of the combined blue-white right triangle - area of the blue triangle = 7.50 - 3
area of the white triangle = 4.50 cm²

area of the rectangle = red + blue + white + green
15 = 2 + 3 + 4.50 + green
15 = 9.50 + green
area of the green triangle = 15 - 9.50
area of the green triangle = 5.50 cm²

in summary of the areas:

area of the rectangle = 15 cm²
red triangle = 2 cm²
blue triangle = 3 cm²
white triangle = 4.50 cm²
green triangle = 5.50 cm²