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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.

See The Solution Submitted by Ady TZIDON    
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Hints/Tips Analytic Answer | Comment 2 of 4 |

I will assume the altitude is the smallest of the four consecutive numbers.  Let the altitude be x, then the sides are x+1, x+2, and x+3.

The area can be expressed by Heron's Formula:
If s=(a+b+c)/2 then the area is sqrt[s*(s-a)*(s-b)*(s-c)]

Applying this to our triangle yields A=(1/4)*sqrt[(3x+6)*(x)*(x+2)*(x+4)].  The area can also be expressed as A=x*y/2 where y is one of x+1, x+2, or x+3.  Then:
(1/4)*sqrt[(3x+6)*(x)*(x+2)*(x+4)] = x*y/2.

Squaring each side and simplifying:
4x^2*y^2 = 3*x*(x+2)^2*(x+4)

Let y=x+2.  Then the equation simplifies to 4x^2 = 3x^2 + 12x, which has a positive integer solution of x=12.  One solution triangle has sides 13,14,15 and altitude 12.

Substituting y=x+1 and y=x+3 did not lead to integer solutions.

  Posted by Brian Smith on 2010-02-15 15:42:36
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