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805+ (Hard)|   Geometry|      
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Bunuel
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An isosceles but not an equilateral triangle has the sum of two of its 29 Apr 2020, 03:37
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An isosceles but not an equilateral triangle has the sum of two of its sides equal to 12. If the lengths of all three sides are integers, how many such isosceles triangles are possible?


A. 6
B. 11
C. 16
D. 17
E. 23


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An isosceles but not an equilateral triangle has the sum of two of its Updated on: 29 Apr 2020, 07:07
Originally posted by GMATWhizTeam on 29 Apr 2020, 04:46.
Last edited by GMATWhizTeam on 29 Apr 2020, 07:07, edited 1 time in total.
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Bunuel
An isosceles but not an equilateral triangle has the sum of two of its sides equal to 12. If the lengths of all three sides are integers, how many such isosceles triangles are possible?


A. 6
B. 11
C. 17
D. 18
E. 23


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Solution:


If the sum of two equal sides is 12 units
    • Since the triangle is not an equilateral triangle third side cannot be 6
    • Third side can be 1, 2, 3, 4, 5, 7, 8, 9, 10, and 11
      o There are ten possibilities
If the sum of two unequal sides is 12 units.
    • There are 6 possibilities in this case (5, 5, 7), (7, 7, 5), (8, 8, 4), (9, 9, 3), (10, 10, 2) and (11, 11, 1)
    • We are not considering the case of (6, 6, 6) because that will make it a equilateral triangle.
    • Possible triangles = 10 + 6 = 16
Hence, the correct answer is Option C
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An isosceles but not an equilateral triangle has the sum of two of its 29 Apr 2020, 05:26
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Bunuel
An isosceles but not an equilateral triangle has the sum of two of its sides equal to 12. If the lengths of all three sides are integers, how many such isosceles triangles are possible?


A. 6
B. 11
C. 17
D. 18
E. 23


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Solution:


If the sum of two equal sides is 12 units
    • Since the triangle is not an equilateral triangle third side cannot be 6
    • Third side can be 7, 8, 9, 10, and 11
      o There are five possibilities
If the sum of two unequal sides is 12 units.
    • There are 6 possibilities in this case (5, 5, 7), (7, 7, 5), (8, 8, 4), (9, 9, 3), (10, 10, 2) and (11, 11, 1)
    • We are not considering the case of (6, 6, 6) because that will make it a equilateral triangle.
    • Possible triangles = 5 + 6 = 11
Hence, the correct answer is Option B
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GMATWhizTeam

If the sum of the two equal sides is 12, why can we not have (6,6,1) (6,6,2) (6,6,3) (6,6,4) and (6,6,5) ?

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An isosceles but not an equilateral triangle has the sum of two of its 29 Apr 2020, 06:15
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you are right.
(6,6,1) (6,6,2) (6,6,3) (6,6,4) and (6,6,5) can be the sides of isosceles triangle.

firas92
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Bunuel
An isosceles but not an equilateral triangle has the sum of two of its sides equal to 12. If the lengths of all three sides are integers, how many such isosceles triangles are possible?


A. 6
B. 11
C. 17
D. 18
E. 23


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Solution:


If the sum of two equal sides is 12 units
    • Since the triangle is not an equilateral triangle third side cannot be 6
    • Third side can be 7, 8, 9, 10, and 11
      o There are five possibilities
If the sum of two unequal sides is 12 units.
    • There are 6 possibilities in this case (5, 5, 7), (7, 7, 5), (8, 8, 4), (9, 9, 3), (10, 10, 2) and (11, 11, 1)
    • We are not considering the case of (6, 6, 6) because that will make it a equilateral triangle.
    • Possible triangles = 5 + 6 = 11
Hence, the correct answer is Option B

GMATWhizTeam

If the sum of the two equal sides is 12, why can we not have (6,6,1) (6,6,2) (6,6,3) (6,6,4) and (6,6,5) ?

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An isosceles but not an equilateral triangle has the sum of two of its 29 Apr 2020, 06:24
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Solution:

While finding the other sides, remember the triangle inequality :

The third side will be less than the sum of the other 2 sides and greater than the difference of the other 2 sides

Case 1: Equal sides are 6,6

(6,6,1) (6,6,2), (6,6,3), (6,6,4), (6,6,5), (6,6,7), (6,6,8) (6,6,9), (6,6,10),(6,6,11) = 10 triangles

Case2: 6,6 are not the equal sides: (11,1,11), (10,2,10), (9,3,9), (8,4,8), (7,5,7), (5,7,5) = 6 triangles

TOTAL = 16 triangles.

Bunuel
An isosceles but not an equilateral triangle has the sum of two of its sides equal to 12. If the lengths of all three sides are integers, how many such isosceles triangles are possible?


A. 6
B. 11
C. 17
D. 18
E. 23


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An isosceles but not an equilateral triangle has the sum of two of its 29 Apr 2020, 06:31
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Solution:

While finding the other sides, remember the triangle inequality :

The third side will be less than the sum of the other 2 sides and greater than the difference of the other 2 sides

Case 1: Equal sides are 6,6

(6,6,1) (6,6,2), (6,6,3), (6,6,4), (6,6,5), (6,6,7), (6,6,8) (6,6,9), (6,6,10),(6,6,11) = 10 triangles

Case2: 6,6 are not the equal sides: (11,1,11), (10,2,10), (9,3,9), (8,4,8), (7,5,7), (5,7,5) = 6 triangles

TOTAL = 16 triangles.

Bunuel
An isosceles but not an equilateral triangle has the sum of two of its sides equal to 12. If the lengths of all three sides are integers, how many such isosceles triangles are possible?


A. 6
B. 11
C. 17
D. 18
E. 23


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My answer was 16 too. Maybe the options need to be corrected?

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An isosceles but not an equilateral triangle has the sum of two of its 29 Apr 2020, 06:32
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firas92
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Solution:

While finding the other sides, remember the triangle inequality :

The third side will be less than the sum of the other 2 sides and greater than the difference of the other 2 sides

Case 1: Equal sides are 6,6

(6,6,1) (6,6,2), (6,6,3), (6,6,4), (6,6,5), (6,6,7), (6,6,8) (6,6,9), (6,6,10),(6,6,11) = 10 triangles

Case2: 6,6 are not the equal sides: (11,1,11), (10,2,10), (9,3,9), (8,4,8), (7,5,7), (5,7,5) = 6 triangles

TOTAL = 16 triangles.

Bunuel
An isosceles but not an equilateral triangle has the sum of two of its sides equal to 12. If the lengths of all three sides are integers, how many such isosceles triangles are possible?


A. 6
B. 11
C. 17
D. 18
E. 23


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My answer was 16 too. Maybe the options need to be corrected?

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_____________________________
Added 16 as an option. Thank you.
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An isosceles but not an equilateral triangle has the sum of two of its 29 Apr 2020, 06:59
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The following 16 triangles all meet the necessary criteria.

1. 6,6,1
2. 6,6,2
3. 6,6,3
4. 6,6,4
5. 6,6,5
6. 6,6,7
7. 6,6,8
8. 6,6,9
9. 6,6,10
10. 6,6,11
11. 5,7,5
12. 7,5,7
13. 8,4,8
14. 9,3,9
15. 10,2,10
16. 11,1,11
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An isosceles but not an equilateral triangle has the sum of two of its 29 Apr 2020, 07:02
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Bunuel
An isosceles but not an equilateral triangle has the sum of two of its sides equal to 12. If the lengths of all three sides are integers, how many such isosceles triangles are possible?


A. 6
B. 11
C. 17
D. 18
E. 23


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Are You Up For the Challenge: 700 Level Questions

Solution:


If the sum of two equal sides is 12 units
    • Since the triangle is not an equilateral triangle third side cannot be 6
    • Third side can be 7, 8, 9, 10, and 11
      o There are five possibilities
If the sum of two unequal sides is 12 units.
    • There are 6 possibilities in this case (5, 5, 7), (7, 7, 5), (8, 8, 4), (9, 9, 3), (10, 10, 2) and (11, 11, 1)
    • We are not considering the case of (6, 6, 6) because that will make it a equilateral triangle.
    • Possible triangles = 5 + 6 = 11
Hence, the correct answer is Option B

GMATWhizTeam

If the sum of the two equal sides is 12, why can we not have (6,6,1) (6,6,2) (6,6,3) (6,6,4) and (6,6,5) ?

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Hey Firas,


You are right, they are possible too. Thanks for pointing it out.

Editing the solution. :)
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An isosceles but not an equilateral triangle has the sum of two of its 02 May 2020, 03:11
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Bunuel
An isosceles but not an equilateral triangle has the sum of two of its sides equal to 12. If the lengths of all three sides are integers, how many such isosceles triangles are possible?


A. 6
B. 11
C. 16
D. 17
E. 23

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Can anyone post a shortcut solution to this?
I mean we aren’t really going to list out all the possibilities here, right?
I mean sure, first case can be listed inside head, (6-6) < Third side < (6 + 6)

But what about the case of “Unequal sides” ?

( 5, 5, 7 )
( 7, 7, 5 )

then just single case for
(8, 8, 4), (9, 9, 3), (10, 10, 2), (11, 11, 1) ?


How will I list out all the cases such as (8, 4, 4) should not be included because
(8 - 4) < 4 < (8 + 4) is illegal in such a short span of time?
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An isosceles but not an equilateral triangle has the sum of two of its 28 Oct 2022, 09:04
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