Question

Consider two circles, each having radius 5 cm, touching each other at a point P. A direct tangent QR is drawn touching one circle at a point Q and the other circle at R. Inside the region PQR inscribed by the two circles and the tangent, a square ABCD is inscribed with its base AB on the tangent and the other side touching the two circles at points D and C, respectively.
Find the area of the square ABCD.

• A. 100 sq. cm
• B. 4 sq. cm
• C. \( 4\sqrt{2} \) sq. cm
• D. None of the other options is correct
• E. 40 sq. cm

Hint:

When dealing with geometric problems involving tangents and inscribed figures, always use symmetry and properties of tangents and circles to simplify the calculations.

Answer 1

The Correct Option is A

Step 1: Understanding the geometry.
Since the radius of each circle is 5 cm and the circles are tangent to each other, the distance between their centers is \( 2 \times 5 = 10 \) cm. The square ABCD is inscribed inside the tangent region formed by the circles and the tangent QR.
Step 2: Use the geometry of the square and the circles.
We know that the length of the side of the square is equal to the distance between the points of tangency on the circles. The tangent segment from a point outside a circle to the point of tangency is perpendicular to the radius at the point of tangency.
Step 3: Calculate the side of the square.
The side of the square is equal to the distance between the centers of the two circles, which is 10 cm. Since the square is inscribed within the region between the circles, the area of the square is \( 10^2 = 100 \, \text{sq. cm} \).
Step 4: Conclusion.
The area of the square ABCD is \( 100 \, \text{sq. cm} \). Therefore, the correct answer is (A).