2020 · Q66
What is the area of the triangle having side lengths y/z + z/x, z/x + x/y, x/y + y/z?
Elementary MathematicsGeometry: Lines, Angles and Triangles
- A((x + y + z)²)/(xyz)
- B√(xyz)/(x + y + z)
- C√(x/y + y/z + z/x)
- D√((xy + yz + zx)/(xyz))
Explanation
Put a = x/y, b = y/z, c = z/x; the sides are b + c, c + a, a + b with semiperimeter s = a + b + c. By Heron, area = √[s(s−a−b)(s−b−c)(s−c−a)] = √[(a+b+c)cab] = √(a + b + c) since abc = 1. That is √(x/y + y/z + z/x).