Which two figures out of the following four have the same area (with same units) ?
(Figure 1: Right triangle with sides 14, 22, 90 degree angle)
(Figure 2: Square with side 12)
(Figure 3: Isosceles triangle with sides 22, 22, 22)
(Figure 4: Circle with radius 7)
Figure 1: Right triangle with legs 14 and 22.
Area₁ = (1/2) * base * height = (1/2) * 14 * 22 = 7 * 22 = 154 sq units.
Figure 2: Square with side 12.
Area₂ = side² = 12² = 144 sq units.
Figure 3: Isosceles triangle with sides 22, 22, 22. This is an equilateral triangle with side length 22.
Area₃ = (√3 / 4) * side² = (√3 / 4) * 22² = (√3 / 4) * 484 = 121√3 sq units.
Using √3 ≈ 1.732, Area₃ ≈ 121 * 1.732 ≈ 209.5 sq units.
Figure 4: Circle with radius 7.
Area₄ = π * radius² = π * 7² = 49π sq units.
Using the approximation π ≈ 22/7 (often used in problems involving multiples of 7):
Area₄ ≈ 49 * (22/7) = 7 * 22 = 154 sq units.
Comparing the areas:
Area₁ = 154
Area₂ = 144
Area₃ ≈ 209.5
Area₄ ≈ 154 (using π ≈ 22/7) or ≈ 153.9 (using π ≈ 3.14)
Figures 1 and 4 have areas 154 and approximately 153.9 or exactly 154 if using π=22/7. The areas are essentially the same.