A triangle has sides of length 12, 15 and k units. What is the length

Question

A triangle has sides of length 12, 15 and k units. What is the length of the longest side?

(1) The perimeter of the triangle is a multiple of 9.
(2) One of the interior angles of the triangle is equal to the sum of the other two interior angles.

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chetan2u · Accepted Answer

Rule :- The third side or k here has to be less than the sum of other sides and more than the difference of other two sides. 
so  15-12<k<15+12.......3<k<27

(1) The perimeter of the triangle is a multiple of 9.
So k+12+15 or k+27 is a multiple of 9. Thus k is a multiple of 9.
Hence k could be 9 or 18
Insuff

(2) One of the interior angles of the triangle is equal to the sum of the other two interior angles.
So let the angles be a, b and c, so a+b+c=180 and a+b=c...c+c=180...c=90
Thus, we have a right angled triangle.
Now the hypotenuse could be k, so  k=\sqrt{12^2+15^2}=\sqrt{369}=3\sqrt{41} 
or the hypotenuse could be 15, so  k=\sqrt{15^2-12^2}=\sqrt{81}=9 
Insuff

Combined
k=9

C

ArunSharma12 · Answer

Given:
3 < K< 27

statement 1:
27 + k = 9*x
k:  
not sufficient

statement 2:
c = a + b
a + b + c =180
triangle is right angled.
case 1: when 15 is largest side
 k^2 + 144 = 225 
 k^2 = 81 0 
K = 9

case 2: when k is largest side
 225 + 144 = k^2 
 369 = k^2 
K is not an integer.
not sufficient

combining both statements,
K = 9.
Ans: C

GMATWhizTeam · Answer

Solution 
 Step 1: Analyse Question Stem 
 • 12, 15 and k units are sides of a triangle.
 o     
• Using the side properties of a triangle, we can write,
 o  15 – 12 < k < 12 + 15 ⟹ 3 < k < 27 ……(i)  
• We need to find the length of the longest side.
 o If we know k we can compare 15 and k and decide which side is the longest.

Step 2: Analyse Statements Independently (And eliminate options) – AD/BCE 
 Statement 1:  The perimeter of the triangle is a multiple of 9.
 • From (i) we have,  12 + 15 + 3 < 12 + 15 + k < 12 + 15 + 27 ⟹ 30 < 9n < 54 , where n is a positive integer.
 o So, n can be either 4 or 5.
  If  n = 4   then  k = 9*4 – 12 – 15 = 9 
 • In this case the longest side is 15  
  If  n = 5  then  k = 9*5 – 12 – 15 = 18 
 • In this case the longest side is 18.
• We are getting two different results.    
Hence, statement 1 is NOT sufficient and we can eliminate answer Options A and D.
 Statement 2:  One of the interior angles of the triangle is equal to the sum of the other two interior angles.
 • Let us assume that the three interior angles of the triangle are a, b, and c.
 o   a+b = c   and  a+b+c = 180  degrees  ⟹2c = 180 ⟹c = 90  degrees.
o So, the given triangle is right angled triangle. However, from this information we cannot decide which one is longest, either one of 15 or k can be longest side.  
Hence, statement 2 is also NOT sufficient and we can eliminate answer Option B.

Step 3: Analyse Statements by combining. 
 • From statement 1: k can be 9 or 18
• From statement 2: Given triangle is a right- angled triangle. So,
 o If k is the longest side then  k^2 = 12^2 + 15^2 ⟹ k = \sqrt{369} 
o If 15 is the longest then  k^2 = 15^2 – 12^2 ⟹ k = 9  
• On combining both statements, we get, k  = 9 
 o Hence, the longest side  = 15   
Thus, the correct answer is  Option C.

Thekingmaker · Answer

With A definitely with figure out clearly insufficient 18,9 are possiblities however when we are going towards we can figure out that it produce a right angled traingle there in we get 9 and 3*(41)^1//2 combining both these we get C therefore IMO C

A triangle has sides of length 12, 15 and k units. What is the length

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A triangle has sides of length 12, 15 and k units. What is the length

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