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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?

  1. 11
  2. 6
  3. 17
  4. 23

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Explanatory Answer

Method of solving this CAT Question from Triangles: Sum of two sides are given, how can we figure out what the third side could be?

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.

Hence, the answer is 17.

Choice C is the correct answer.

 

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