Question

use the diagram to answer the questions on the right:
triangle with vertices b, a, c. angle at c is right angle. side ba is 11, side bc is x, angle at b is 60 degrees.

Explanation:

Step1: Identify triangle type

Triangle \( ABC \) is right - angled at \( C \), with \( \angle B = 60^\circ \) and hypotenuse \( AB = 11 \), side \( BC=x \). In a right - triangle, \( \cos\theta=\frac{\text{adjacent}}{\text{hypotenuse}} \). Here, \( \theta = 60^\circ \), adjacent side to \( \angle B \) is \( BC=x \), hypotenuse \( AB = 11 \). So \( \cos(60^\circ)=\frac{x}{11} \).

Step2: Solve for \( x \)

We know that \( \cos(60^\circ)=\frac{1}{2} \). Substituting into the equation \( \frac{1}{2}=\frac{x}{11} \), we can solve for \( x \) by cross - multiplying: \( x = 11\times\frac{1}{2}=\frac{11}{2}=5.5 \).

Answer:

\( x = \frac{11}{2} \) (or \( 5.5 \))