[tex]\bf \textit{arc's length}\\\\ s=\cfrac{\theta \pi r}{180}~~ \begin{cases} r=radius\\ \theta =angle~in\\ \qquad degrees\\[-0.5em] \hrulefill\\ r=6.5\\ s=5.7 \end{cases}\implies 5.7=\cfrac{\theta \pi (6.5)}{180}\implies 1026=6.5\pi \theta \\\\\\ \cfrac{1026}{6.5\pi }=\theta \implies \stackrel{\textit{rounded up, using }\pi =3.14}{50.27=\theta }[/tex]
The measure of the angle at the centre of the circle made by the arc of length 5.7 cm that has a radius of 6.5 cm is 50.27°.
Any smooth curve connecting two points is called an arc. The arc length is the measurement of how long an arc is. The length of an arc is given by the formula,
[tex]\rm{ Length\ of\ an\ Arc = 2\times \pi \times(radius)\times\dfrac{\theta}{360} = 2\pi r \times \dfrac{\theta}{2\pi}[/tex]
where
θ is the angle, that arc creates at the centre of the circle in degree.
As we know that the length of an arc is given by the formula,
[tex]\rm{ Length\ of\ an\ Arc = 2\pi r \times \dfrac{\theta}{360^o}[/tex]
Given the length of the arc is 5.7 cm, while the radius of the circle is 6.5cm, therefore, the angle made by the arc at the centre of the circle is,
[tex]5.7 = 2\pi (6.5) \times \dfrac{\theta}{360^o}\\\\[/tex]
θ = 50.27°
Hence, the measure of the angle at the centre of the circle made by the arc of length 5.7 cm that has a radius of 6.5 cm is 50.27°.
Learn more about the Length of an Arc:
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