Question

* A chord PQ of a circle of radius 5 cm subtends an angle of 70° at the centre. Calculate the following: a) b) c) the length of the chord PQ the length of the arc PQ the perimeters of sector and segment.​

Answer 1

Check the picture below.

so let's get the chord using the pythagorean theorem hmmm using sine

`\sin(35^o )=\cfrac{\stackrel{opposite}{x}}{\underset{hypotenuse}{5}}\implies 5\sin(35^o )=x\implies 2.87\approx x~\hfill \underset{ PQ }{\stackrel{ 2.87+2.87 }{\approx \text{\LARGE 5.74}}}`

now let's get the arc

`\textit{arc's length}\\\\ s = \cfrac{\theta \pi r}{180} ~~ \begin{cases} r=radius\\ \theta =\stackrel{degrees}{angle}\\[-0.5em] \hrulefill\\ \theta =70\\ r=5 \end{cases}\implies s=\cfrac{(70)\pi (5)}{180}\implies s\approx \text{\LARGE 6.11}`

and the perimeters, keeping in mind that for the sector is just the arc plus the radii, and for the segment is simply the arc plus the chord.

`\stackrel{ \textit{sector's perimeter} }{5+5+6.11 ~~ \approx ~~} \text{\LARGE 16.11}\hspace{5em}\stackrel{ \textit{segment's perimeter} }{5.74+6.11 ~~ \approx ~~} \text{\LARGE 11.85}`