Question

in the diagram, $mangle c = 35$, $moverarc{bd} = 50$, $overline{de} cong overline{ba}$ and the radius of the circle is 10 cm. what is the arc length of $overarc{de}$ to the nearest hundredth of a centimeter?

(options: 15.86 cm, 16.86 cm, 18.34 cm, 16.58 cm)

Explanation:

Step1: Relate angle to arc measures

Let $m\overset{\frown}{DE}=m\overset{\frown}{BA}=x$. The formula for the measure of an angle formed by two secants outside a circle is $m\angle C=\frac{1}{2}(m\overset{\frown}{DE}-m\overset{\frown}{BD})$.
Substitute known values: $35=\frac{1}{2}(x - 50)$

Step2: Solve for $x$

Multiply both sides by 2: $70=x - 50$
Add 50 to both sides: $x=70 + 50=120$
So $m\overset{\frown}{DE}=120^\circ$

Step3: Calculate arc length

Arc length formula: $L=\frac{\theta}{360}\times2\pi r$, where $\theta=120^\circ$, $r=10$ cm
Substitute values: $L=\frac{120}{360}\times2\pi\times10$
Simplify: $L=\frac{1}{3}\times20\pi=\frac{20\pi}{3}\approx16.58$ cm

Answer:

16.58 cm