Area Ratios With Midpoints
In the picture above, a rectangle's midpoints are connected by line segments to form a rhombus inscribed within the rectangle. Similarly, the midpoints of the previously constructed rhombus are connected to form a smaller rectangle inscribed within the rhombus, and the smaller rectangle's midpoints are also connected to form a smaller rhombus inscribed within it. Furthermore, a vertical line segment whose endpoints are two midpoints of the large rectangle is drawn connecting two midpoints of the small rectangle.
What is the ratio of the blue shaded area to the orange shaded area? Submit the answer in decimal form if necessary.
e.g - Ratio = 5:3
The answer is 1.
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As shown above, dividing the rectangle into 32 congruent triangles makes the problem a lot simpler. Using the image above, we can count the number of blue triangles and orange triangles to give us the ratio of their total areas as such:
Blue Area = 10 △
Orange Area = 10 △
∴ Blue Area = Orange Area ⇒ 1 : 1 ⇒ 1 1 ⇒ 1