The perimeter of a rhombus is 100 meters and one of its diagonals is 14 meters long. What is the length of the other diagonal?

Question

The perimeter of a rhombus is 100 meters and one of its diagonals is 14 meters long. What is the length of the other diagonal?

1. A. 48 meters (Correct answer)
2. B. 40 meters
3. C. 42 meters
4. D. 50 meters

Correct Answer

Option A — 48 meters

Detailed Solution & Explanation

The correct answer is 48 meters.

Shortcut Trick

• A rhombus has four equal sides, so side $= \dfrac{100}{4} = 25$ m.
• Its diagonals bisect each other at right angles, forming four right triangles with legs $\dfrac{d_1}{2}$, $\dfrac{d_2}{2}$ and hypotenuse 25.
• $7^2 + \left(\dfrac{d_2}{2}\right)^2 = 25^2 \Rightarrow \left(\dfrac{d_2}{2}\right)^2 = 625 - 49 = 576 \Rightarrow \dfrac{d_2}{2} = 24$
• $d_2 = \textbf{48}$ m.

Alternate Method

• Using the identity $d_1^2 + d_2^2 = 4a^2$ for a rhombus of side $a$:
  • $14^2 + d_2^2 = 4(25)^2$
  • $196 + d_2^2 = 2500$
  • $d_2^2 = 2304 \Rightarrow d_2 = \textbf{48}$ m

Additional Information

• The half-diagonals and side form the 7–24–25 Pythagorean triple, which is what makes the arithmetic come out whole. Recognising 7-24-25 alongside 3-4-5 and 5-12-13 saves real time.
• Area of a rhombus $= \dfrac{1}{2} d_1 d_2 = \dfrac{1}{2}(14)(48) = 336$ m².
• All rhombus properties follow from it being a parallelogram with equal sides: opposite angles equal, diagonals bisecting the angles, and diagonals perpendicular — the last being unique to the rhombus among parallelograms.