Question

mary is given the diagram below, showing an angle rotation of 120°. the arc length of the sector is 40 cm.
statement 1
the radius of the circle, to the nearest centimetre, is 19 cm.
statement 2
an equivalent angle rotation is \\( \frac { 4 \pi } { 3 } \\).
statement 3
if the arc length on this circle increases to 80 cm, then the central angle must be 240°.
statement 4
mary can determine the radius of the circle by dividing the given angle by the arc length.

1. mathematically prove or disprove each of the statements above. 8 marks

Explanation:

Statement 1

Step1: Use the arc - length formula \(l = r\theta\) (where \(\theta\) is in radians).

Given \(l = 40\) cm and \(\theta=120^{\circ}=\frac{120\pi}{180}=\frac{2\pi}{3}\) radians.
From \(l = r\theta\), we can solve for \(r\) as \(r=\frac{l}{\theta}\).

Step2: Substitute the values of \(l\) and \(\theta\).

\(r=\frac{40}{\frac{2\pi}{3}}=\frac{40\times3}{2\pi}=\frac{60}{\pi}\approx19\) cm.
So, Statement 1 is correct.

Statement 2

Step1: Convert \(120^{\circ}\) to radians.

Using the conversion formula \(\theta_{rad}=\frac{\pi}{180}\times\theta_{deg}\), for \(\theta = 120^{\circ}\), \(\theta=\frac{120\pi}{180}=\frac{2\pi}{3}\) radians.
So, Statement 2 is incorrect.

Statement 3

Step1: Use the arc - length formula \(l = r\theta\) (where \(r\) is the radius and \(\theta\) is the central angle in radians).

We know \(l = r\theta\), originally \(l = 40\), \(r=\frac{60}{\pi}\), \(\theta=\frac{2\pi}{3}\). If \(l = 80\), and \(r=\frac{60}{\pi}\), then \(\theta=\frac{l}{r}=\frac{80}{\frac{60}{\pi}}=\frac{4\pi}{3}\) radians.
Convert \(\frac{4\pi}{3}\) radians to degrees: \(\theta_{deg}=\frac{180}{\pi}\times\frac{4\pi}{3}=240^{\circ}\).
So, Statement 3 is correct.

Statement 4

Step1: Recall the formula for radius \(r=\frac{l}{\theta}\) (where \(\theta\) is in radians).

The formula for radius is \(r = \frac{l}{\theta}\), not dividing the given angle by the arc length.
So, Statement 4 is incorrect.

Answer:

• Statement 1: Correct.
• Statement 2: Incorrect.
• Statement 3: Correct.
• Statement 4: Incorrect.