online cat preparation time CAT Preparation - Geometry Question 05

Question: How many isosceles triangles with integer sides are possible such that sum of two of the side is 12?

A) 11
B) 6
C) 17
D) 23

DETAILED SOLUTION:

Join OE and OD.
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.
11 + 6 = 17 possibilities totally.

Answer choice (C)

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online cat preparation time CAT Preparation - Geometry Question 05
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