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

Explanation:

Step1: Recall the arc - length formula

The formula for the arc - length \(l\) of a sector with radius \(r\) and central angle \(\theta\) (in degrees) is \(l=\frac{\theta}{360}\times2\pi r\).

Step2: Analyze Statement 1

Given \(r = 40\mathrm{cm}\) and \(\theta=120^{\circ}\). Substitute into the formula: \(l=\frac{120}{360}\times2\pi\times40=\frac{1}{3}\times80\pi=\frac{80\pi}{3}\approx\frac{80\times3.14}{3}\approx83.73\mathrm{cm}
eq19\mathrm{cm}\). So, Statement 1 is False.

Step3: Analyze Statement 2

Convert \(\theta = 120^{\circ}\) to radians. We know that \(\theta\) (in radians) \(=\frac{\pi}{180}\times\theta\) (in degrees). So, \(\theta=\frac{\pi}{180}\times120=\frac{2\pi}{3}\). The arc - length formula \(l = r\theta\) (when \(\theta\) is in radians). If \(r = 2\), then \(l=2\times\frac{2\pi}{3}=\frac{4\pi}{3}\). So, Statement 2 is True.

Step4: Analyze Statement 3

Using \(l=\frac{\theta}{360}\times2\pi r\). Let the original \(r_1 = 40\), \(\theta_1 = 120^{\circ}\), \(l_1=\frac{120}{360}\times2\pi\times40\). New \(r_2 = 80\), \(l_2=\frac{\theta_2}{360}\times2\pi\times80\). If \(l_1 = l_2\), then \(\frac{120}{360}\times2\pi\times40=\frac{\theta_2}{360}\times2\pi\times80\). Cancel out \(\frac{2\pi}{360}\) on both sides. We get \(120\times40=\theta_2\times80\), \(\theta_2 = 60^{\circ}
eq240^{\circ}\). So, Statement 3 is False.

Step5: Analyze Statement 4

The formula \(l=\frac{\theta}{360}\times2\pi r\) requires knowing both \(\theta\) and \(r\) to calculate \(l\). Dividing the radius by a number without knowing the central angle does not directly give the arc - length. For example, if \(r\) is divided by \(k\) (\(k>0\)), \(l=\frac{\theta}{360}\times2\pi\times\frac{r}{k}\) (if \(\theta\) is fixed). But we don't know \(\theta\) here. So, Statement 4 is False.

Answer:

Statement 1: False; Statement 2: True; Statement 3: False; Statement 4: False.