CAT question from geometry - triangles. level of difficulty: Medium

1. Geometry

2. Geometry
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

3. Geometry
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?
How many isosceles triangles with integer sides are possible such
that sum of two of the side is 12?

4. Geometry
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.
How many isosceles triangles with integer sides are possible such
that sum of two of the side is 12?

5. Geometry
 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?
How many isosceles triangles with integer sides are possible such
that sum of two of the side is 12?

6. Geometry
 Now 2 unequal sides adding to 12. This could be
1 + 11, 2 + 10, 3 + 9, 4 + 8 or 5 + 7
Isosceles triangles with the above combination:
1, 11, 11  1, 1, 11 x
2, 10, 10  2, 2, 10 x
3, 9, 9  3, 3, 9 x
4, 8, 8  4, 4, 8 x
5, 7, 7  5, 5, 7 
How many isosceles triangles with integer sides are possible such
that sum of two of the side is 12?

7. Geometry
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)
How many isosceles triangles with integer sides are possible such
that sum of two of the side is 12?

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