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The three sides and the height (Posted on 2010-02-15) Difficulty: 4 of 5
The sides and height of a triangle are 4 consecutive integers. Evaluate the triangle's area.

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Solution Solution | Comment 3 of 4 |
Given the three sides and height of a triangle as consecutive integers: x, (x+1), (x+2) and (x+3).
The height of the triangle must be x or x+1.

If the height = x, the sides of the triangle are (x+1), (x+2) and (x+3). If the height = (x+1), then the sides of the triangle are x, (x+2) and (x+3).

Using the Pythagorean equations, one can derive cubic equations for three sets of triangles where (x+3) is the hypotenuse of the larger triangle with height and leg of length h, and where the hypotenuse of the smaller triangle, if extended, would intersect the base of the larger triangle.

For h = x with the shorter hypotenuse (x+1), the cubic equation would be: x3 - 8x2 - 44x - 48 = 0. Solving the cubic, the one real root is equal to 12.

For h = x with the shorter hypotenuse (x+2), the cubic equation would be:  x3 - 16x2 - 56x - 48 = 0. Solving the cubic, the one real root is a non-integer: ~19.06875, and therefore not a solution.

For h = (x+1) with the shorter hypotenuse (x+2), the cubic equation would be:  2x3 + 3x2 - 4x - 8 = 0. Solving the cubic, the one real root is a non-integer: ~1.52657, and therefore not a solution.

Therefore the only solution is where x=12, as the height of the triangle, and the length of the base, (x+2) = 14. The area of this triangle is 1/2*(14)*(12) = 84.

 

Edited on February 16, 2010, 1:53 am
  Posted by Dej Mar on 2010-02-16 00:56:01

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