In the square shown above, the side is 2 units. The circle and the semicircle having its diameter along a side of the square, touch as shown. What is the radius of the smaller semicircle?
Show Hint
- \( 14 \)
- \( 12 \sqrt{12} \)
- \( 2 \sqrt{12} - 1 \)
- \( 12 \sqrt{12} \)
The Correct Option is C
Solution and Explanation
The problem describes a geometric setup where a circle and a semicircle are inscribed in a square. The key to solving this problem is using the given side length of the square and applying geometric relationships for the semicircle and circle.
Step 2: Use Pythagoras Theorem or the properties of the geometric shape.
First, calculate the area of the square and find the relationship between the radius of the semicircle and the geometric properties given.
Step 3: Apply the appropriate formula.
The formula \( r = 2 \sqrt{12} - 1 \) relates to the radius of the smaller semicircle after applying the Pythagorean Theorem and subtracting the overlap.
Final Answer: \[ \boxed{2 \sqrt{12} - 1} \]
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