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# Triangle ABC has sides of lengths 7, 13 and x, where x is the length

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Joined: 02 Sep 2009
Posts: 55277
Triangle ABC has sides of lengths 7, 13 and x, where x is the length  [#permalink]

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15 Aug 2017, 02:20
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Difficulty:

45% (medium)

Question Stats:

65% (01:32) correct 35% (01:53) wrong based on 46 sessions

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Triangle ABC has sides of lengths 7, 13 and x, where x is the length of the longest side. If x is the square of an integer, the perimeter of ∆ ABC is equal to which of the following?

(A) 24
(B) 29
(C) 36
(D) 45
(E) 49

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Re: Triangle ABC has sides of lengths 7, 13 and x, where x is the length  [#permalink]

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15 Aug 2017, 02:34
C
Side 6<x <20
Perfect square can be 9 or 16
But x is longest so >13
Hence 16.
Perimeter 36.
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Re: Triangle ABC has sides of lengths 7, 13 and x, where x is the length  [#permalink]

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15 Aug 2017, 02:47
Bunuel wrote:
Triangle ABC has sides of lengths 7, 13 and x, where x is the length of the longest side. If x is the square of an integer, the perimeter of ∆ ABC is equal to which of the following?

(A) 24
(B) 29
(C) 36
(D) 45
(E) 49

The third side(x) which is a square of an integer can be 16 or 9
But a triangle with sides 7,9 and 13 cannot be possible x must be the larger side.
The perimeter could be 7+13+16 which is 36(Option C)
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Re: Triangle ABC has sides of lengths 7, 13 and x, where x is the length  [#permalink]

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18 Aug 2017, 09:12
Bunuel wrote:
Triangle ABC has sides of lengths 7, 13 and x, where x is the length of the longest side. If x is the square of an integer, the perimeter of ∆ ABC is equal to which of the following?

(A) 24
(B) 29
(C) 36
(D) 45
(E) 49

The perimeter of the triangle is 7 + 13 + x = 20 + x. Let’s analyze each answer choice.

A) 24

If the perimeter is 24, then:

20 + x = 24 → x = 4

Since 4 cannot be the longest side, answer A is not correct.

B) 29

If the perimeter is 29, then :

20 + x = 29 → x = 9

Since 9 cannot be the longest side, answer B is not correct.

C) 36

If the perimeter is 24, then:

20 + x = 36 → x = 16

Since 16 is a perfect square and it is larger than 7 and 13, 16 can be the longest side and 36 can be the perimeter.

Alternate Solution:

Since x is the longer side, x > 13. Furthermore, by the triangle inequality, x should be less than the sum of the other two sides; therefore x < 7 + 13 = 20. That is, 13 < x < 20. Looking for a perfect square integer between 13 and 20, we see that 16 is the only choice. Therefore, the perimeter of the triangle is 7 + 13 + 16 = 36.

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Re: Triangle ABC has sides of lengths 7, 13 and x, where x is the length   [#permalink] 18 Aug 2017, 09:12
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