Question

IF NR = 8 ft, what is the length of arc NMP. Round to the nearest tenth.

Answer 1

Check the picture below.

since chords NQ and MP cross the center of the circle at R, that means that those two chords are diametrical chords and the angles made by both are vertical angles and thus twin angles, namely both are 18° as you see in the picture, so the angle NMP in magenta is really 162° + 18° + 18° = 198°, and we know the radius NR is 8.

`\textit{arc's length}\\\\ s=\cfrac{r\pi \theta }{180}~~ \begin{cases} r=radius\\ \theta =\stackrel{degrees}{angle}\\[-0.5em] \hrulefill\\ r=8\\ \theta =198 \end{cases}\implies s=\cfrac{(8)\pi (198)}{180}\implies s\approx 27.6`

Answer 2

Since the radius of circle R is 8 feet, the length of arc NMP is equal to 27.7 feet.

How to calculate the length of the arc?

In Mathematics and Geometry, the arc length formed by a circle can be calculated by using the following equation (formula):

Arc length = 2πr × θ/360

Where:

• r represents the radius of a circle.
• θ represents the central angle.

First of all, we would determine the measure of the two vertical angles that forms the central angle of the arc;

m∠MRQ ≅ measure of arc MQ = 162°

m∠MRN = 180 - m∠MRQ

m∠MRN = 180 - 162

m∠MRN = 18°

Hence, the central angle subtended by arc NMP is given by;

Central angle, θ = m∠MRN + 180°

Central angle, θ = 18° + 180°

Central angle, θ = 198°

Now, we can determine length of arc NMP;

Arc length = 2πr × θ/360

Arc length = 2 × π × 8 × 198/360

Arc length = 69696/2520

Arc length = 27.7 feet.