Question

in circle c, cd = 10 and the length of de = 8π. find m∠dce.

Explanation:

Step1: Recall the arc - length formula

The formula for the length of an arc \(l=\frac{n\pi r}{180}\), where \(l\) is the arc length, \(n\) is the central angle in degrees, and \(r\) is the radius of the circle.
Given that \(CD = r=10\) (radius of the circle) and \(l = 8\pi\) (length of arc \(\overset{\frown}{DE}\)).
Substitute the values into the formula: \(8\pi=\frac{n\pi\times10}{180}\).

Step2: Solve for \(n\)

First, cancel out \(\pi\) from both sides of the equation \(8\pi=\frac{n\pi\times10}{180}\). We get \(8=\frac{10n}{180}\).
Then, cross - multiply: \(10n=8\times180\).
So, \(10n = 1440\).
Divide both sides by 10: \(n=\frac{1440}{10}=144\).

Answer:

\(144\)