What equilateral triangle would have the same area as a triangle with side 6, 8 and 10?
did you notice that 6,8, and 10 forms a right-angled triangle (6^2 + 8^2 = 10^2)
so the area of the given triangle - (1/2)(6)(8) = 24
the area of an equilateral triangle with sides x is
(1/2)x^2 sin60° = √3/4 x^2
√3/4 x^2 = 24
x^2 = 96/√3
x = √(96/√3) = appr 7.44
To find the equilateral triangle with the same area as the given triangle with side lengths 6, 8, and 10, we can use the formula for the area of a triangle.
Given the side lengths 6, 8, and 10, we can use Heron's formula to find the area of this triangle. Heron's formula states that the area, A, of a triangle with side lengths a, b, and c is given by:
A = √(s(s-a)(s-b)(s-c))
where s is the semi-perimeter of the triangle, given by:
s = (a + b + c) / 2
For the given triangle, substituting the side lengths, we have:
s = (6 + 8 + 10) / 2 = 24 / 2 = 12
Now we can calculate the area using Heron's formula:
A = √(12(12-6)(12-8)(12-10))
= √(12 * 6 * 4 * 2)
= √(1152)
≈ 33.941
Now that we have the area of the given triangle, we can find the side length of the equilateral triangle with the same area.
The formula for the area of an equilateral triangle, A_eq, is given by:
A_eq = (sqrt(3)/4) * s_eq^2
where s_eq is the side length of the equilateral triangle.
To find the side length, we rearrange the formula:
s_eq = √(4 * A_eq / sqrt(3))
Substituting the area of the given triangle into the formula:
s_eq = √(4 * 33.941 / sqrt(3))
= √(135.764 / sqrt(3))
= √(135.764 * sqrt(3) / (sqrt(3) * sqrt(3)))
= √(135.764 * sqrt(3) / 3)
≈ √(45.255 * sqrt(3))
Therefore, the side length of the equilateral triangle that has the same area as the original triangle is approximately √(45.255 * sqrt(3)).