Question

If \[a = 9\], \[b = 8\] and \[c = x\] satisfies the equation \[3\cos C = 2\]. Then what is the value of \[x\]?
A. 5
B. 6
C. 4
D. 7

Answer 1

Hint: Here, the given values are the length of the sides of a triangle. So, we will use the law of cosines for the angle \[C\]. Substitute the values from the given information and solve it to get the required answer.



Formula Used:Law of cosines for the angle \[C\]: \[\cos C = \dfrac{{{a^2} + b{}^2 - {c^2}}}{{2ab}}\]



Complete step by step solution:Given:
 \[a = 9\], \[b = 8\] and \[c = x\] are the sides of the given triangle.
The above values satisfy the equation \[3\cos C = 2\].

Let’s apply the law of cosines for the angle \[C\].
We get,
\[\cos C = \dfrac{{{a^2} + b{}^2 - {c^2}}}{{2ab}}\]
Substitute the values from the given equations.
\[\dfrac{2}{3} = \dfrac{{{9^2} + 8{}^2 - {x^2}}}{{2 \times 9 \times 8}}\]
\[ \Rightarrow \dfrac{2}{3} = \dfrac{{81 + 64 - {x^2}}}{{144}}\]
\[ \Rightarrow \dfrac{2}{3} = \dfrac{{145 - {x^2}}}{{144}}\]
\[ \Rightarrow 96 = 145 - {x^2}\]
\[ \Rightarrow {x^2} = 49 \]
Take the square root on both sides.
\[x = \pm 7 \]
We know that the length of the sides of a triangle cannot be negative.
So, \[x = 7 \]



Option ‘D’ is correct



Note: The law of cosines is the ratio of the lengths of the sides of a triangle with respect to the cosine of its angle.
The law states that the square of one side is equal to the sum of the squares of the other sides minus twice the product of these sides and the cosine of the intermediate angle.