Q.5 In the following diagram, the point R is the center of the circle. The lines PQ
and ZV are tangential to the circle. The relation among the areas of the squares,
PXWR, RUVZ and SPQT is
(A) Area of SPQT = Area of RUVZ = Area of PXWR
(B) Area of SPQT = Area of PXWR – Area of RUVZ
(C) Area of PXWR = Area of SPQT – Area of RUVZ
(D) Area of PXWR = Area of RUVZ – Area of SPQT

Tangential Geometry Puzzle – Comparing Areas of Three Squares Touching a Circle

Introduction

This famous reasoning puzzle compares the areas of three squares drawn around a circle. The circle has center R, and two lines PQ and ZV are tangent to it. From these tangents, three different squares — PXWR, SPQT, and RUVZ — are constructed.

The key question: Which square is largest? Or are their areas equal?

Understanding how tangents relate to radii makes solving this puzzle easy.

Step-by-Step Solution

Given Information

• R is the center of the circle
• PQ and ZV are tangents
• Squares are: PXWR, SPQT, RUVZ

Key Geometry Principle

If a line touches a circle and a line from the point of tangency is drawn to the center, that line is perpendicular to the tangent.

Therefore:

• RQ ⊥ PQ
• RZ ⊥ ZV

Why Are All Three Square Sides Equal?

Each square uses: side = radius of the circle

Consider square SPQT:

• Q lies on the circle
• PQ is tangent at Q
• RQ is radius
• Tangent-radius perpendicular means Q is a vertex of a square whose side equals RQ = radius

Similarly:

• Square RUVZ uses the radius at point Z
• Square PXWR uses the radius rotated to point P/W

Thus, each square has side length = radius of the circle, so:

Area = (side)² = r²

Therefore:

Area_SPQT = Area_PXWR = Area_RUVZ

Conclusion

All three squares are built outward from the circle using tangents at radius endpoints. Since each square has the same side length, all have the same area.

Therefore: Correct Answer: (A) Area of SPQT = Area of RUVZ = Area of PXWR

Why the Other Options Are Wrong