Let a, b,c be the sides of a triangle and c be the larger side.
A triangle can be formed by the sides a,b and c only if,
[tex]\begin{gathered} a+b>c \\ b+c>a \\ c+a>b \end{gathered}[/tex]Then, if
[tex]\begin{gathered} c^2=a^2+b^2 \\ \end{gathered}[/tex]The triangle is right angled.
If,
[tex]c^2>a^2+b^2[/tex]The triangle is obtuse.
If,
[tex]c^2The triangle is acute.Consider sides 2, 4,5.
The sum of any two sides is greater than the third side. So, a triangle can be formed by the sides 2,4, and 5. 5 is the length of the largest side.
[tex]\begin{gathered} 5^2>2^2+4^2 \\ 25>4+16 \end{gathered}[/tex]Hence, 2,4,5 are the sides of an obtuse angled triangle.
Consider sides 3, 4, 5.
The sum of any two sides is greater than the third side. So, a triangle can be formed by the sides 3, 4, and 5. 5 is the length of the largest side.
[tex]\begin{gathered} 5^2=3^2+4^2 \\ 25=9+16 \end{gathered}[/tex]Hence, 3,4 and 5 are the sides of a right angled triangle.
Consider sides 6,7,8.
The sum of any two sides is greater than the third side. So, a triangle can be formed by the sides 6, 7, and 8. 8 is the length of the largest side.
[tex]\begin{gathered} 8^2<6^2+7^2 \\ 64<36+49 \end{gathered}[/tex]Hence, 6,7 and 8 are the sides of an acute angled triangle.
Consider sides 7,9,15.
The sum of any two sides is greater than the third side. So, a triangle can be formed by the sides 7, 9, and 15. 15 is the length of the largest side.
[tex]15^2>7^2+9^2[/tex]Hence, 7, 9and 15 are the sides of an obtuse angled triangle.
Consider sides 3,3,10.
The sum of any two sides is not greater than the third side. So, a triangle cannot be formed by the sides 3,3 and 10.
