C
Side 6<x <20
Perfect square can be 9 or 16
But x is longest so >13
Hence 16.
Perimeter 36.

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.

Answer: C
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