If the area of a square is equal to the area of an equilateral triangle, the…
2023
If the area of a square is equal to the area of an equilateral triangle, the ratio of the side of the square and the side of the equilateral triangle is?
Answer: E. None of the above — ConceptThe area of a square with side s is s2. The area of an equilateral triangle with side t is (√3/4)t2. For equal areas, the two formulas are equated and…
- A.
1:1
- B.
2:1
- C.
\(\sqrt{3}:2\)
- D.
More than one of the above
- E.
None of the above
Attempted by 14 students.
Show answer & explanation
Correct answer: E
Concept
The area of a square with side s is s2. The area of an equilateral triangle with side t is (√3/4)t2. For equal areas, the two formulas are equated and the positive side ratio is obtained.
Application
Let s be the side of the square and t be the side of the equilateral triangle.
Equal areas give s2 = (√3/4)t2.
Dividing by t2 gives (s/t)2 = √3/4.
Since side lengths are positive, s/t = √(√3/4) = 31/4/2. Hence s:t = 31/4:2.
Cross-check
Take s = 31/4 and t = 2. The square area is √3.
The triangle area is (√3/4) × 22 = (√3/4) × 4 = √3, confirming equal areas.
Result
The required ratio is 31/4:2. It is not among the three listed numerical ratios, so the applicable choice is “None of the above.”