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.
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.
To determine whether it is possible for the common ratio, r, to be 1/5 or not possible for it to be 2 in a geometric progression (G.P.) formed by the lengths of the sides of a triangle, we need to consider the properties of triangles and G.P.s.
In a triangle, the length of any side must be greater than the positive difference between the lengths of the other two sides. This is commonly known as the Triangle Inequality Theorem.
Let's analyze the two possible values of r:
1) r = 1/5:
In this case, the G.P. with a common ratio of 1/5 would be: 10 cm, 10/5 = 2 cm, 2/5 = 0.4 cm, and so on.
The triangle would have side lengths of 10 cm, 2 cm, and 0.4 cm. These lengths satisfy the Triangle Inequality Theorem since:
- 10 cm > |2 cm - 0.4 cm| = 1.6 cm
- 2 cm > |10 cm - 0.4 cm| = 9.6 cm
- 0.4 cm > |10 cm - 2 cm| = 8 cm
Therefore, it is possible for r to be 1/5 because the resulting side lengths create a valid triangle.
2) r = 2:
In this case, the G.P. with a common ratio of 2 would be: 10 cm, 20 cm, 40 cm, and so on.
The triangle would have side lengths of 10 cm, 20 cm, and 40 cm. However, these lengths do not satisfy the Triangle Inequality Theorem since:
- 10 cm < |20 cm - 40 cm| = 20 cm
- 20 cm < |10 cm - 40 cm| = 30 cm
- 40 cm < |10 cm - 20 cm| = 10 cm
Therefore, it is not possible for r to be 2 because the resulting side lengths do not form a valid triangle.
In summary, it is possible for r to be 1/5 because the resulting side lengths follow the Triangle Inequality Theorem, while it is not possible for r to be 2 since the resulting side lengths violate the Triangle Inequality Theorem.
To understand why it is possible for r to be 1/5 but not possible for r to be 2 in this scenario, let's break down the conditions of a geometric progression (G.P.) and consider the triangle's side lengths.
In a G.P., each term is obtained by multiplying the previous term by a constant ratio, denoted by "r" in this case.
Given that the shortest side of the triangle is 10 cm, we can denote it as the first term of the G.P: a₁ = 10 cm.
Now, let's analyze the suitability of different values for the common ratio "r."
If r = 1/5:
In this case, the terms of the G.P. would be:
a₁ = 10 cm
a₂ = a₁ * r = 10 cm * (1/5) = 2 cm
a₃ = a₂ * r = 2 cm * (1/5) = 0.4 cm
As you can see, the terms in this G.P. would decrease significantly, indicating that the side lengths of the triangle would also decrease. So, it is indeed possible for r to be 1/5.
If r = 2:
In this case, the terms of the G.P. would be:
a₁ = 10 cm
a₂ = a₁ * r = 10 cm * 2 = 20 cm
a₃ = a₂ * r = 20 cm * 2 = 40 cm
Here, the terms in the G.P. increase, indicating an increase in side lengths. However, if we assume 10 cm is the shortest side, having a subsequent side length (e.g., 20 cm, 40 cm) greater than the sum of the other two sides violates the triangle inequality theorem. According to the theorem, the sum of the lengths of any two sides of a triangle must be greater than the length of the third side.
Since having r = 2 leads to side lengths violating the triangle inequality theorem, it is not possible for r to be 2 in this scenario.
Hence, r can be 1/5 but not 2 in this particular case.