Question

Suppose the three sides of a triangle are of lengths a, b, and c respectively. Further suppose that a, b, and c are positive integers with a sb sc. a) Given that c = 7, find the number of different triangles that can be formed by assigning appropriate values to a and b. b) Given that c = 8, find the number of different triangles that can be formed by assigning appropriate values to a and b. c) Given that c = 9, find the number of different triangles that can be formed by assigning appropriate values to a and b.

Answer 1

Based on the triangle inequality theorem, when c=7, 8, and 9 respectively, there can be 21, 28, and 36 distinct triangles formed respectively.

Step-by-step explanation:

The subject of the question is about different types of triangles that can be formed with sides of certain lengths. This is a problem related to combinatorics and geometry.

The issue at hand involves the triangle inequality theorem, which states that the sum of the lengths of any two sides of a triangle must be greater than the length of the third side.

a) When c = 7, the maximum possible value for a and b would be 6. Thus, the number of distinct triangles you could form are [(6-1)+1] * [(6-1+1)/2] = 21.

b) When c = 8, the maximum possible values for a and b would be 7. Therefore, the number of distinct triangles you could form are [(7-1)+1] * [(7-1+1)/2] = 28.

c) When c = 9, the maximum possible values for a and b would be 8. Therefore, the number of distinct triangles you could form are [(8-1)+1] * [(8-1+1)/2] = 36.

Learn more about Triangle Inequality Theorem