The sides and height of a triangle are 4 consecutive integers.
Evaluate the triangle's area.
It would seem that the height (altitude) would divide the triangle into two Pythagorean triangles such that one of the sides would be considered the base. That base would be the sum of two integers.
My spreadsheet has columns headed:
A A+1 A+2 A+ 3 S Area1 Area2 Area3 Area4
"A" lists incremental values while the three adjacent columns increment that value by "1"
Since triangular area is calculated as Base*Height/2 I assumed that any side could be the base.
Area1 is half the product of (A+1)*A,
Area2 is half the product of (A+2)*A and
Area3 is half the product of (A+3)*A.
Area4 utilises the Heron formula using the increments of A as the side lengths with S being half of their sum.
Highlighting my first data row I copied the formula downwards.
At A=12 I got the following data:
A A+1 A+2 A+ 3 S Area1 Area2 Area3 Area4
12 13 14 15 21 136.5
84 157.5
84Investigating this I found that the
131415 triangle was composed of two Pythagorean triangles 51312 and 91512; 9+5=14 and
12 is the altitude.

Posted by brianjn
on 20100216 01:41:56 