Two sides of a triangle are 12 cm and 7 cm; find the range for the length of its third side.

According to the triangle inequality theorem, sum of the lengths of any two sides of a triangle is always greater than the third side.

Two sides of the triangle are 12 cm and 7 cm. Let c be the third side of triangle.

⇒ 12 + 7 > c

⇒ 19 > c

And, 12 + c > 7

⇒ c > -5 (side cannot be negative)

And, 7 + c > 12

⇒ c > 5

5 < c < 19 cm

Hence, the range for the length of the third side is between 5 cm and 19 cm.