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

Solution

Step 1: Analyse Question Stem

• 12, 15 and k units are sides of a triangle.

o [We don’t know whether k is integer or not.]
• 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

• 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.

• 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.

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 \(= 15\)