The length of the sides of a triangle form a G.P. Given that the shortest side of the triangle Is 10 cm and the common ratio is r. explain why it is possible for r to be 1:5 but not possible for r to be 2.

Answer 1

In a geometric progression (G.P.), each term is obtained by multiplying the previous term by a constant ratio. If the shortest side of the triangle is 10 cm, and the common ratio is 1:5, this means that the second side of the triangle would be the shortest side multiplied by the common ratio:

Second side = 10 cm * 1:5 = 10 cm * 1.5 = 15 cm

Similarly, the third side of the triangle would be the second side multiplied by the common ratio:

Third side = 15 cm * 1:5 = 15 cm * 1.5 = 22.5 cm

Therefore, with a common ratio of 1:5, the side lengths of the triangle would be 10 cm, 15 cm, and 22.5 cm, which is a valid triangle.

However, if the common ratio were 2, the second side of the triangle would be the shortest side multiplied by the common ratio:

Second side = 10 cm * 2 = 20 cm

In this case, the second side would be longer than the sum of the other two sides (10 cm + 20 cm = 30 cm), violating the Triangle Inequality Theorem. According to this theorem, the sum of the lengths of any two sides of a triangle must be greater than the length of the third side. Consequently, it would be impossible to form a triangle with side lengths of 10 cm, 20 cm, and any other length.

Therefore, while a common ratio of 1:5 is possible in a G.P. for the side lengths of a triangle, a common ratio of 2 is not possible.