3,4,5 ; 5,12,13 ; 9,12,15 are the possible measures of the triangle's sides.
Explanation:
Let us consider a triangle having three sides,
For a normal triangle all the three sides will be equal , but as the given values are not equal it should be a right angle triangle.
The area of right angle triangle is given by pythagoras theorem.
(Area of 1st side)² + (Area of 2nd side)² = (Area of 3rd side)²
Now as we have given different values we have to choose which series suits a triangle
i.) 3,4,5
By applying theorem,
1st side² + 2nd side² = 3rd side²
3² + 4² = 5²
9 + 16 = 25
25=25
ii.) 5, 12, 13
similarly as above
5² + 12² = 13²
169 = 169
iii.)9, 12 ,15
9² + 12² = 15²
81 + 144 = 225
225 = 225
Therefore these are the values of series which best suits and forms right angled triangle.