The sides of a triangle are in arithmetic progression and its area is 3/5th the area of an equilateral triangle with the same perimeter.
Find the ratio of the sides of the triangle.
Heron's formula states that the area of any triangle is √p(p-a)(p-b)(p-c), where p is the semiperimeter, and a, b, and c are the sides. Let b = x, a = x - d, and c = x + d; then the area is √(3x/2)(x/2 + d)(x/2)(x/2 - d). Simplifying, this is x/2 * √3x^2/4 - d^2.
The area of a triangle with side length s is s^2 * (√3)/4. s = x in this case, so:
x/2 * √(3x^2/4 - d^2) = x^2 * (√3)/4 * 3/5
√(3x^2/4 - d^2) = 3x/10 * √3
3x^2/4 - d^2 = 27x^2/100
12/25 * x^2 = d^2
d = 2x/5 * √2
The ratio is then x - 2x/5 * √2 : x : x + 2x/5 * √2. Cancelling out the x's:
(5 - 2√2)/5 : 1 : (5 + 2√2)/5
(Honestly, did you guys think it would come out with rationals? The area of an equilateral triangle always involves √3 in some way.)
Solution
