Question

The perimeter of a triangle is not greater than 12 and the lengths of sides are natural numbers.

How many such kinds of triangles is it possible? Find the probability of randomly chosen triangle is isosceles but not equilateral.

Answer 1

To find the number of triangles with a perimeter not greater than 12 and natural number side lengths, we can use the concept of partitions. There are 12 different possible combinations, so there are 12 different triangles that satisfy the given conditions. The probability of randomly choosing an isosceles triangle but not an equilateral triangle is 1/12.

Step-by-step explanation:

To find the number of triangles with a perimeter not greater than 12 and natural number side lengths, we can use the concept of partitions. Let's list all the possible partitions of the number 12 into three parts:

1 + 1 + 10

1 + 2 + 9

1 + 3 + 8

1 + 4 + 7

1 + 5 + 6

2 + 2 + 8

2 + 3 + 7

2 + 4 + 6

2 + 5 + 5

3 + 3 + 6

3 + 4 + 5

4 + 4 + 4

There are 12 different possible combinations, so there are 12 different triangles that satisfy the given conditions.

To find the probability of randomly choosing an isosceles triangle but not an equilateral triangle, we need to count how many of the 12 triangles are isosceles but not equilateral. In this case, only the last combination, 4 + 4 + 4, satisfies the condition. Therefore, the probability is 1/12.

Answer 2

3 triangles

Step-by-step explanation:

Perimeter of triangle = a + b + c

Given that :

P = 12

and a, b, c are natural numbers

Let :

Side A = a

Side B = b

Side C = 12 - (a + b)

Side A + side B > side C - - - (condition 1)

a + b > 12 - (a + b)

a + b > 12 - a - b

a + a + b + b > 12

2a + 2b > 12

2(a + b) > 12

a + b > 6

Side A - side B < side C

a - b < 12 - (a + b)

a - b + a + b < 12

2a < 12

a < 6

b < 6 (arbitrary point)

Going by the Constraint above :

The only three possibilities are :

(2, 5, 5)

(3, 4, 5)

(4, 4, 4)

Total number of triangle = 3

Equilateral triangle (all 3 sides equal) = (4, 4, 4) = 1

Isosceles triangle (only 2 sides equal) = (2, 5, 5) = 1