Integer Triangle is an area of
Geometry and Number theory in which a lot of work has been added in the past
decade. A lot of questions on Integer triangles are part of Recreational
Mathematics now. Besides many examiners have started including such questions
in different aptitude tests.
An integer triangle is
a triangle whose all sides have lengths that are integers.
There are a few general properties
for an integer triangle.
Finding no. of Integer Triangles when
Perimeter is given:
Suppose we have to find the no. of Integer Triangles when
the perimeter is given as equal to 20.
Here we use the triangular inequality and realize that the
longest side of such triangle can be 9 and we can tabulate the data as below:
|
N0.
|
Longest Side a
|
Side b
|
Side c
|
Perimeter
|
|
1
|
9
|
9
|
2
|
20
|
|
2
|
9
|
8
|
3
|
20
|
|
3
|
9
|
7
|
4
|
20
|
|
4
|
9
|
6
|
5
|
20
|
|
5
|
8
|
8
|
4
|
20
|
|
6
|
8
|
7
|
5
|
20
|
|
7
|
8
|
6
|
6
|
20
|
|
8
|
7
|
7
|
6
|
20
|
Now it can be seen that calculating in this manner is very
cumbersome. If the perimeter given is higher it will take much longer time.
However we can use a simple formula for such calculation.
Now there may be two cases; the perimeter may be even or odd
Case 1: when the perimeter, say p, is even:
The no. of triangles is given by the formula nint (p2/48)
Where, nint means nearest Integer Function.
For example, nint (1.6) is equal to 2 and nint (3.2) =3.
( Additionally half-integers are always rounded to even
numbers For example, nint(1.5)=2, nint(2.5)=2, nint(3.5)=4, nint(4.5)=4, etc.)
For our example of perimeter equal to 20; we can calculate
in this manner
nint (20x20/48)= nint (8.33) =8
Case 2: when the perimeter,
say p, is odd:
The no. of triangles is given by the formula nint {(p+3)2/48}
Finding no. of Integer Triangles when
longest side is given:
Case 1: When the longest, say a, is even.
No. of Integer triangles = a(a+1)
Case 2: When the longest, say a, is odd.
No. of Integer triangles = a2
( To be Continued)