A Captivating Geometry Challenge Involving Circles and Squares
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Chapter 1: Exploring the Geometry of Circles
In this geometry challenge, we are presented with three circles, designated as A, B, and C, each having a radius of 1. Circles A and B touch at a single point, while Circle C is tangent to the midpoint of the line segment connecting A and B. The question posed is: Can you determine the area of the shaded region?
For those eager to engage with this problem, I suggest pausing here, grabbing some paper and a pen, and attempting to solve it before continuing to the solution!
This video titled "Fun Geometry Challenge" provides a visual and interactive approach to tackling similar geometry problems.
Solution
There are several methods to find the answer, but we will explore a straightforward yet elegant solution in this discussion. We will designate point D as the midpoint of the line segment AB, and points E and F will represent the intersections of Circle C with Circles A and B, respectively.
Next, we identify point G, which lies directly opposite point D on Circle C. By connecting the points E, D, F, and G, we can form a square whose diagonal is equal to the diameter of the circle, which measures 2.
The elegance of geometry is revealed as we manipulate shapes and space. Observe in the diagram below how parts of the shaded area can be rearranged to fill the unshaded sections of the square seamlessly.
This indicates that the area of the shaded region corresponds to that of the square. Utilizing the Pythagorean Theorem, we find the side length of the square to be √2, leading to the area calculation:
A = (√2)² = 2
And there you have it—our result!
How fascinating is that?
What were your thoughts while solving this problem? Feel free to share in the comments; I'm eager to hear your insights!
The second video, "Cool Geometry Problems used to Teach Geometry?" delves into various methods to teach and understand geometry concepts, enhancing your problem-solving skills.
Math Puzzles: A Collection of Engaging Challenges
Discover an array of math puzzles on Medium, encompassing Algebra, Geometry, Calculus, Number Theory, and more. Don't hesitate to share this collection with your friends!
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Happy solving,
Bella