Chapter 1: The Geometry Challenge

Have you ever come across intricate floor tile designs in cathedrals? This geometry puzzle invites you to explore such patterns and challenges you to determine the side length of the inner square. Interestingly, this design exhibits four axes of symmetry, adding another layer of complexity!

Before you proceed, I encourage you to take a moment to grab some paper and a pen to work through the problem. Once you feel prepared, continue for the solution!

Here, we present an important geometric observation that will guide our approach to the problem.

Section 1.1: Analyzing the Diagram

To tackle this geometry puzzle effectively, let’s begin by annotating the diagram.

• Designate point O as the center of the circle.
• Let P represent one intersection of the two quarter circles.
• Point Q will be the center of one of the quarter circles.
• Finally, let R be the vertex of the square that lies on OQ, as depicted.

Next, we turn our attention to triangle OPQ, where we can observe a right angle at point P, and both QP and OP measure 1 unit.

Subsection 1.1.1: Applying Pythagoras' Theorem

Utilizing Pythagoras' Theorem, we can calculate the length of OQ:

OQ = √(1² + 1²) = √2

Observe the line segment QO, which is composed of segments QR and RO. Notably, the length of QR equals the radius of each quarter circle, which is 1 unit. Consequently, we can conclude that OR = √2 - 1.

This means the diagonal of the square equals double the length of OR:

Diagonal of the square = 2(√2 - 1)

Now, let's analyze a right triangle where the hypotenuse represents the square's diagonal. Again, we apply Pythagoras' Theorem, letting x represent the side length of the square:

x² + x² = (2(√2 - 1))², leading us to find x = 2 - √2

Thus, we arrive at our final answer.

Chapter 2: Explore Further with Geometry Challenges

To further stimulate your mathematical mind, check out these engaging videos:

In the first video, CAN YOU SOLVE THIS GEOMETRY CHALLENGE?, viewers are invited to tackle a challenging geometry problem that will test their skills and creativity.

The second video, Fun Geometry Challenge, offers another opportunity to engage with geometry in an entertaining and educational manner.

What were your thoughts while solving this puzzle? I’m excited to hear your insights in the comments below!

To enrich your math skills, consider saving and sharing this list featuring the best math puzzles on Medium, covering a variety of topics from algebra to calculus.

Math Puzzles

Discover a collection of the best math puzzles available on Medium, spanning various branches of mathematics including Algebra, Geometry, and Number Theory.

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