The question is from CAT Geometry - Triangles. It discusses about isosceles triangles with integer sides. CAT Geometry questions are heavily tested in CAT exam. It tests our understanding of our knowledge about isosceles triangles.
Question 2: How many isosceles triangles with integer sides are possible such that sum of two of the side is 12?
Two possibilities: Two equal sides could add up to 12 or sum of 2 unequal sides = 12.
i.e. Sum of 2 equal sides = 12
Sum of 2 unequal sides = 12
=> If sum of two equal sides were 12, sides of the triangle should be 6, 6, x.
What are the values x can take?
x could range from 1 to 11.
11 integer values exist.
Now 2 unequal sides adding to 12. This could be 1 + 11, 2 + 10, 3 + 9, 4 + 8 or 5 + 7.
How many isosceles triangles are possible with the above combinations?
Isosceles triangles with the above combination:
1, 11, 11   ☑      1, 1, 11    ☒
2, 10, 10   ☑      2, 2, 10    ☒
3, 9, 9       ☑      3, 3, 9       ☒
4, 8, 8       ☑      4, 4, 8       ☒
5, 7, 7       ☑      5, 5, 7       ☑
6 possibilities. Triplets such as (1, 1, 11), (2, 2, 10), etc are eliminated as sum of the two smaller values is less than the largest value. These cannot form a triangle.
The question is " How many isosceles triangles with integer sides are possible such that sum of two of the side is 12?"
11 + 6 = 17 possibilities totally.
Choice C is the correct answer.
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