Question

The circle in the figure below has a radius of r and center at C. The distance from A to B is x, the distance from A to D is y, and the length of arc BD is s. Redraw the figure, label it as indicated, and then solve the problem.

If A = 45°, s = 18, and r = 14, find y. (Round your answer to the nearest whole number.)

Answer 1

Answer: y=19

Step-by-step explanation:

The figure of the triangle is attached and we have the following data:

$A=45\°$

$s=18$ is the length of arc

$r=14$ is the radius

This problem can be solved by the Law of Sines:

$\frac{sin A}{r}=\frac{sin C}{y}=\frac{sin D}{r+x}$

In order to find $C$, we will use the formula of the length of arc:

$s=\frac{2 \pi r C}{360\°}$

Then: $C=\frac{360\° s}{2 \pi r}$

$C=\frac{360\° (18)}{2 \pi (14)}=73.66\°$

Returning to the Law of Sines:

$\frac{sin A}{r}=\frac{sin C}{y}$

Finding $y$:

$y=\frac{sin C r}{sin A}$

$y=\frac{sin (73.66\°) 14}{sin (45\°)}$

Finally:

$y=18.99 \approx 19$