if the perimeter a triangle is 36,what is the lenght of the longest side?

Question

perimeter of triangle is 36 cm

Answer 1

The sides could be 11cm, 12cm and 13cm because 11+12+13=36cm.

Also, the triangle rule is obeyed: 11+12>13, 11+13>12, 12+13>11, so the longest side is 13cm.

More generally if the sides are a, b and c cms long then:

(1) a+b+c=36; (2) a+b-c=C², (3) a+c-b=B², (4) b+c-a=A², where A, B and C are numbers we need to find as well as a, b and c.

(1)+(2): 2a+2b=C²+36, a+b=½C²+18, b=½C²+18-a; (2)+(3): a=½(B²+C²), so b=½C²+18-½(B²+C²​)=18-½B²;

c=36-a-b=36-½(B²+C²)-(18-½B²)=18-½C². Also, b+c-a=A², so 18-½B²+18-½C²-½(B²+C²)=A², so A²+B²+C²=36.

Let A, B and C be even numbers: A=2x, B=2y, C=2z, A²+B²+C²=4(x²+y²+z²)=36, x²+y²+z²=9, so, for example, x=1, y=z=2, A=2, B=C=4.

a=½(B²+C²)=16; b=18-½B²=10; c=18-½C²=10. a+b+c=16+10+10=36cm, and the triangle rule is obeyed. In this example the longest side measures 16cm (base of an isosceles triangle).

There are other examples