Question

I really need this question someone please help ​

Answer 1

Answer:

$\approx 15.9$

Step-by-step explanation:

The length of an arc with measure $\theta$  and radius $r$ is given by $\ell_{arc}=2r\pi\cdot \frac{\theta}{360}$. From the figure, we know that the radius of arc ADC is 4, but we don't know the measure of the arc. Since there are 360 degrees in a circle, the measure of arc ADC is equal to the measure of the arc formed by $\angle AOC$ subtracted from 360. The measure of the arc formed by $\angle AOC$ consists of two congruent angles, $\angle AOB$ and $\angle COB$. To find them, we can use basic trigonometry for a right triangle, since by definition, tangents intersect a circle at a right angle.

In any right triangle, the cosine of an angle is equal to its adjacent side divided by the hypotenuse, or longest side, of the triangle.

We have:

$\cos \angle AOB=\cos \angle COB=\frac{4}{10},\\\angle AOB=\arccos(\frac{4}{10})=66.42182152^{\circ}$

Therefore, $\angle AOC=2\cdot 66.42182152=132.84364304^{\circ}$

The measure of the central angle of $\widehat{ADC}$ must then be $360-132.84364304=227.15635696^{\circ}$

Thus, the length of $\widehat{ADC}$ is equal to:

$\ell_{\widehat{ADC}}=2\cdot 4\cdot \pi \cdot \frac{227.15635696}{360},\\\ell_{\widehat{ADC}}=15.8585053832\approx \boxed{15.9}$ (three significant figures as requested by question).