Question 10 of 14intermediate🔍 AnalyzeShort Answer3 marks

Which has greater area—an equilateral triangle or a square of the same sidelength as the triangle? Which has greater area—two identical equilateral triangles together or a square of the same sidelength as the triangle? Give reasons.

Correct Answer

Part 1: Equilateral triangle vs Square (same sidelength)

Given: Sidelength = s

Formula: Area of equilateral triangle = (3\sqrt{3}/4) × s²

Formula: Area of square = s²

Calculation: Area of equilateral triangle ≈\approx 0.433s² Area of square = s²

Comparison: Since 1 > 0.433, s² > (3\sqrt{3}/4) × s²

∴ The square has greater area than the equilateral triangle.

Part 2: Two equilateral triangles vs Square

Given: Two identical equilateral triangles with sidelength s

Calculation: Combined area = 2 × (3\sqrt{3}/4) × s² = (3\sqrt{3}/2) × s² ≈\approx 0.866s² Area of square = s²

Comparison: Since s² > 0.866s²

∴ The square still has greater area than two equilateral triangles combined.

Exercise: Figure it Out (Section 4) | Q: 9 | (Page Page 17)
For More Understanding

Explanation

The chapter emphasizes comparing areas using formulas. Area of equilateral triangle = (3\sqrt{3}/4) × side², while area of square = side². Comparing these values shows the square has greater area.

Solution Steps

  1. Step 1: Area of equilateral triangle with side s = (3\sqrt{3}/4) × s².

  2. Step 2: Area of square with side s = s².

  3. Step 3: Compare: s² > (3\sqrt{3}/4) × s² since 3\sqrt{3}/4 ≈\approx 0.433 < 1.

  4. Step 4: Two triangles: Area = 2 × (3\sqrt{3}/4) × s² = (3\sqrt{3}/2) × s² ≈\approx 0.866s².

  5. Step 5: Compare: s² > 0.866s², so square still has greater area.

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