Question

Which set of numbers may represent the lengths of the sides of a triangle? (A) {2,5,9} (B) {6,6,7} (C) {6,4,2} (D) {7,8,1}

Answer 1

(B) {6,6,7}

Explanation:

A criterion to determine if each triplet represents a triangle is the Law of Cosine, which states that:

`a^(2) = b^(2)+c^(2)-2\cdot b \cdot c \cdot \cos \theta`

Where
`a`,
`b` and
`c` are sides of the triangle and
`\theta` is the angle opposite to side
`a`. Now, let is clear the cosine function:

`2\cdot a \cdot b\cdot \cos \theta = b^(2)+c^(2)-a^(2)`

`\cos \theta = (b^(2)+c^(2)-a^(2))/(2\cdot b \cdot c)`

Cosine is a bounded function between -1 and 1, a triplet corresponds to a triangle if and only if result is located between upper and lower bounds. Now let is evaluate each triplet:

a)
`a = 2`,
`b = 5`,
`c = 9`

`\cos \theta =(5^(2)+9^(2)-2^(2))/(2\cdot (5)\cdot (9))`

`\cos \theta = 1.133` (Absurd)

The triplet does not represent a triangle.

b)
`a = 6`,
`b = 6`,
`c = 7`

`\cos \theta =(6^(2)+7^(2)-6^(2))/(2\cdot (6)\cdot (7))`

`\cos \theta = 0.583` (Reasonable)

The triplet represents a triangle.

c)
`a = 6`,
`b = 4`,
`c = 2`

`\cos \theta = (4^(2)+2^(2)-6^(2))/(2\cdot (4)\cdot (2))`

`\cos \theta = -1` (Absurd)

The triplet does not represent a triangle, but a straight line.

d)
`a = 7`,
`b = 8`,
`c = 1`

`\cos \theta = (8^(2)+1^(2)-7^(2))/(2\cdot (8)\cdot (1))`

`\cos \theta = 1` (Absurd)

The triplet does not represent a triangle, but a straight line.

Hence, the correct answer is B.