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

 See The Solution Submitted by Ady TZIDON No Rating

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 A Spreadsheet Perspective Comment 4 of 4 |
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.

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    84

Investigating this I found that the 13-14-15 triangle was composed of two Pythagorean triangles 5-13-12 and 9-15-12; 9+5=14 and 12 is the altitude.
 Posted by brianjn on 2010-02-16 01:41:56

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