Question

in circle ( n ), ( mangle onp = 144^{circ} ) and the area of the shaded sector ( = 40pi ). find the length of ( overline{no} ).

Explanation:

Step1: Recall the formula for the area of a sector

The formula for the area of a sector of a circle is \(A=\frac{\theta}{360^{\circ}}\times\pi r^{2}\), where \(\theta\) is the central - angle of the sector and \(r\) is the radius of the circle. Here, \(\theta = 144^{\circ}\) and \(A = 40\pi\), and \(r=NO\).

Step2: Substitute the values into the formula

Substitute \(\theta = 144^{\circ}\) and \(A = 40\pi\) into \(A=\frac{\theta}{360^{\circ}}\times\pi r^{2}\). We get \(40\pi=\frac{144^{\circ}}{360^{\circ}}\times\pi r^{2}\).
First, simplify \(\frac{144^{\circ}}{360^{\circ}}=\frac{2}{5}\). So the equation becomes \(40\pi=\frac{2}{5}\pi r^{2}\).

Step3: Solve for \(r^{2}\)

Divide both sides of the equation \(40\pi=\frac{2}{5}\pi r^{2}\) by \(\pi\) (since \(\pi
eq0\)). We have \(40=\frac{2}{5}r^{2}\).
Then multiply both sides by \(\frac{5}{2}\) to isolate \(r^{2}\). \(r^{2}=40\times\frac{5}{2}\).
\(40\times\frac{5}{2}=100\).

Step4: Solve for \(r\)

Take the square root of both sides. Since \(r>0\) (as it represents the length of a radius), \(r = \sqrt{100}\).

Answer:

\(NO = 10\)