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.
Answer: C
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