Twelve equal squares are placed to fit in a rectangle of diagonal 5 cm. There are three rows containing four squares each. No gaps are left between adjacent squares. What is the area of each square?
- (a) 5/7 sq cm
- (b) 7/5 sq cm
- (c) 1 sq cm ✓ UPSC's answer
- (d) 25/12 sq cm
Why the answer is (c)
• Let the side length of each square be $s$ cm.
• The rectangle is formed by 3 rows and 4 columns of squares, so its width is $4s$ and its height is $3s$.
• The diagonal of the rectangle is given as 5 cm.
• Using the Pythagorean theorem: $(4s)^2 + (3s)^2 = 5^2$.
• This simplifies to $16s^2 + 9s^2 = 25$, which gives $25s^2 = 25$.
• Solving for $s^2$ yields $s^2 = 1$, which is the area of each square.
Why the other options are wrong
- (a) 5/7 sq cm
- This option incorrectly assumes the area is derived from a non-integer side length or misapplies the diagonal formula.
- (b) 7/5 sq cm
- This option is the reciprocal of the correct area and does not satisfy the Pythagorean relationship for the given dimensions.
- (d) 25/12 sq cm
- This option incorrectly calculates the area by dividing the total diagonal squared by the number of squares without accounting for the aspect ratio.
Asked in the CSAT Paper II of the UPSC Civil Services Preliminary Examination 2018, held on 3 June 2018. Question and answer key: Union Public Service Commission. Explanation: UPSC Answer Check.