Question

The area of an equilateral triangle described on one side of the square is equal to how many times the area of the equilateral triangle described on one of its diagonal?

• A. $\frac{1}{4}$ times
  
• B. $\frac{1}{2}$ times
• C. $\frac{1}{8}$ times
• D. $\frac{1}{16}$ times

Answer 1

The correct option is B $\frac{1}{2}$ times

Let the side of a square be $a$

Length of diagonal $=\sqrt{a^2+a^2}=a\sqrt{2}$

Ratio $=\frac{\text{Area of equilateral triangle with edge}a}{\text{Area of equilateral triangle with edge}a\sqrt{2}}$

$=\frac{\frac{\sqrt{3}}{4}{a}^2}{\frac{\sqrt{3}}{4}{\left(a\sqrt{2}\right)}^2}$

$=\frac{1}{2}$

Hence, the area of an equilateral triangle described on one side of a square is equal to half the area of the equilateral triangle described on one of its diagonals