Answer:
(B) {6,6,7}
Step-by-step explanation:
A criterion to determine if each triplet represents a triangle is the Law of Cosine, which states that:
[tex]a^{2} = b^{2}+c^{2}-2\cdot b \cdot c \cdot \cos \theta[/tex]
Where [tex]a[/tex], [tex]b[/tex] and [tex]c[/tex] are sides of the triangle and [tex]\theta[/tex] is the angle opposite to side [tex]a[/tex]. Now, let is clear the cosine function:
[tex]2\cdot a \cdot b\cdot \cos \theta = b^{2}+c^{2}-a^{2}[/tex]
[tex]\cos \theta = \frac{b^{2}+c^{2}-a^{2}}{2\cdot b \cdot c}[/tex]
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) [tex]a = 2[/tex], [tex]b = 5[/tex], [tex]c = 9[/tex]
[tex]\cos \theta =\frac{5^{2}+9^{2}-2^{2}}{2\cdot (5)\cdot (9)}[/tex]
[tex]\cos \theta = 1.133[/tex] (Absurd)
The triplet does not represent a triangle.
b) [tex]a = 6[/tex], [tex]b = 6[/tex], [tex]c = 7[/tex]
[tex]\cos \theta =\frac{6^{2}+7^{2}-6^{2}}{2\cdot (6)\cdot (7)}[/tex]
[tex]\cos \theta = 0.583[/tex] (Reasonable)
The triplet represents a triangle.
c) [tex]a = 6[/tex], [tex]b = 4[/tex], [tex]c = 2[/tex]
[tex]\cos \theta = \frac{4^{2}+2^{2}-6^{2}}{2\cdot (4)\cdot (2)}[/tex]
[tex]\cos \theta = -1[/tex] (Absurd)
The triplet does not represent a triangle, but a straight line.
d) [tex]a = 7[/tex], [tex]b = 8[/tex], [tex]c = 1[/tex]
[tex]\cos \theta = \frac{8^{2}+1^{2}-7^{2}}{2\cdot (8)\cdot (1)}[/tex]
[tex]\cos \theta = 1[/tex] (Absurd)
The triplet does not represent a triangle, but a straight line.
Hence, the correct answer is B.
