Hey guys, I've got a maths problem!!?

Question

A triangle is scalene, does not contain a right angle, has integer length sides, and its area is an integer.

Find the triangle with these properties with the least perimeter.

(By the way I found a formula to work out the area of a triangle without knowing the height - sqrt[s(s-a)(s-b)(s-c)] where s is the semi-perimeter a+b+c/2.)

Thanks, a best answer will be voted for!!

Answer 1

You can solve the problem applying the cited formula (Heron's formula) for the area:
A = sqrt[s(s-a)(s-b)(s-c)], s = (a+b+c)/2(semi-perimeter), proceeding as follows:
let x = s - a, y = s - b, z = s - c, then s = x + y + z,
a = s - x, b = s - y, c = s - z the question becomes: for what x, y, z /natural numbers, x+y+z=s/ the product s*x*y*z = A^2 is a perfect square. The latter is impossible if s is a prime number because x, y and z are less than s, so let's check s = 4, 6, 8, 9, 10, 12, 14, 15, 16,... etc. The combinations are quite few, summarized in the following table:

s = x + y + z; sides and area; type
--------------------------------------------------------------------
4 do not exist
6 = 1 + 2 + 3; 3, 4, 5 and 6; Pythagorean, sorry!
8 = 2 + 3 + 3; 5, 5, 6 and 12; isosceles, sorry!
9 = 1 + 4 + 4; 5, 5, 8 and 12; isosceles, sorry!
10 do not exist
12 = 2 + 4 + 6; 6, 8, 10 and 24; Pythagorean, sorry!
14 do not exist
15 = 2 + 3 + 10; 5, 12, 13 and 30; Pythagorean, sorry!
16 = 4 + 6 + 6; 10, 10, 12 and 48; isosceles, sorry!
16 = 1 + 3 + 12; 4, 13, 15 and 24; finally we got it!

So the required scalene non-Pythagorean triangle with the least perimeter is 4, 13 and 15, A=24.

Along with the original problem we did more:
(1) If we allow the triangle to be isosceles than the optimal solution would be 5, 5, 6, A=12 /2 Pythagorean 3, 4, 5 triangles, joined along 4-side/. The second best isosceles 5, 5, 8 is also composed of 2 Pythagorean 3, 4, 5 triangles, joined along 3-side;
(2) If we allow right triangles the champion would be 3, 4, 5, A=6. The same is optimal among ALL triangles with integer sides and integer area.

P.S. If You like arithmetic, I'll suggest to try Yourself on the following: two of the above mentioned triangles: 6,8,10 and 5,12,13 have the interesting property: the area is numerically equal to the perimeter. What triangles with integer sides and integer area have the same property?
The problem has 5 solutions.

Answer 2

The three sides are : 4, 13 and 15 units in length.
The area is 24 square units.
The perimeter is 32 units, being the least perimeter.

Answer 3

any 3 sides forming a triangle has an area
but since you are asking for minimum perimeter and sides are integer
I choose sides as 1 , 1 and 1 . equilateral triangle of side =1
2s = 1+ 1 + 1
s = 3/2
area =sqr [ s (s - a)(s - b)(s - c)
area = sqr [ 3/2 (3/2 - 1)(3/2 - 1)(3/2 - 1) = sqr [ ( 3/2 ) (1/2)^3 ]
area = sqr (3 / 16 ) = sqr (3) / 4

Answer 4

sqrt[s(s-a)(s-b)(s-c)]=integer=k
min a+b+c=?

p=2s=a+b+c

k^2=(s-a)(s-b)(s-c)
=(s^2-bs-as+ab)(s-c)
=s^3-bs^2-as^2+abs-cs^2+bcs+acs-abc
=s^3-(a+b+c)s^2+(ab+bc+ac)s-abc
=s^3-(2s)s^2+(ab+bc+ac)s-abc