Question

what is the length of $overline{bc}$? round to the nearest tenth.
diagram: right triangle with right angle at c, vertex a with $65^circ$ angle, hypotenuse $ab = 16$ cm, side $bc$ labeled $x$. options: $6.8$ cm, $7.5$ cm, $14.6$ cm (partially visible)

Explanation:

Step1: Identify the trigonometric ratio

In right triangle \(ABC\) (right - angled at \(C\)), we know the hypotenuse \(AB = 16\) cm and we want to find the length of \(BC\) (opposite to angle \(A = 65^{\circ}\)). The sine of an angle in a right triangle is defined as \(\sin(\theta)=\frac{\text{opposite}}{\text{hypotenuse}}\). So, \(\sin(65^{\circ})=\frac{BC}{AB}\).

Step2: Solve for \(BC\)

We know that \(AB = 16\) cm and \(\theta=65^{\circ}\). From \(\sin(65^{\circ})=\frac{BC}{16}\), we can re - arrange the formula to get \(BC = 16\times\sin(65^{\circ})\).
We calculate \(\sin(65^{\circ})\approx0.9063\). Then \(BC = 16\times0.9063 = 14.5008\approx14.5\) cm (the third option seems to have a typo, but based on calculation, it should be around 14.5 cm).

Answer:

14.5 cm (assuming the third option is 14.5 cm)