A difficult math question

Question

So I have a question I just can't solve myself. The problem goes like this:

The ratios of the sides of the Triangle are given: 5a=3b=2c, i.e. 5 times the length of the shortest side corresponds to 3 times the length of the second shortest side and 2 times the length of the longest side.

Answer 1

S/M/L

6/10//15

if Smallest is 6, Largest is 15

as 5x6 = 30 = 2x15

and Middle sized side is 10

as 5x6 =30 = 3x10

sides 6,10, 15 are the smallest integer sides of a similar triangle

other similar triangles are

6z, 10z and 15z where z = any positive number

such as

12, 20 and 30

12x5 = 60 = 2x30 = 3x20

or

18, 30, and 45

they're always in a ratio of 6/10/15

If you know 3 sides of one triangle

and 1 side of a 2nd similar triangle you can calculate the missing 2 sides

if one triangle is 6/10/15

and one side of a 2nd similar triangle is 24 where 24 corresponds to side 6

then the other missing two sides are 40 and 60. 4x10 and 4x15 since 24=4x6

Answer 2

A cool way to think of this is 5 x 3 x 2 = 30, and if you want all three expressions to equal each other then a = 3 x 2, b = 5 x 2 and c = 5 x 3.

So the triangle is 6 / 10 / 15 or any multiple of those numbers, like 12 / 20 / 30 or 18 / 30 / 45 or many more. (Actually, there are an infinite number of possibilities.)