Answer:
Length of arc QR is
9.9 cm
Step-by-step explanation:
Given that circle P, i.e. center is point P.
QS is diameter with length 20 cm.
Given that RP is the radius with
![\angle RPS = 123^\circ](https://tex.z-dn.net/?f=%5Cangle%20RPS%20%20%3D%20123%5E%5Ccirc)
To find length of arc QR = ?
<u>Solution:</u>
Arc QR subtends the
on center P.
So, we need to find the angle
to find the length of arc QR.
QS is the diameter so ![\angle QPS = 180^\circ](https://tex.z-dn.net/?f=%5Cangle%20QPS%20%3D%20180%5E%5Ccirc)
![\angle QPS = 180^\circ = \angle QPR +\angle RPS\\\Rightarrow 180^\circ = \angle QPR +123^\circ\\\Rightarrow \angle QPR = 57^\circ](https://tex.z-dn.net/?f=%5Cangle%20QPS%20%3D%20180%5E%5Ccirc%20%3D%20%5Cangle%20QPR%20%2B%5Cangle%20RPS%5C%5C%5CRightarrow%20180%5E%5Ccirc%20%3D%20%5Cangle%20QPR%20%2B123%5E%5Ccirc%5C%5C%5CRightarrow%20%5Cangle%20QPR%20%3D%2057%5E%5Ccirc)
Converting in radians,
![\angle QPR = 57^\circ \times \dfrac{\pi}{180} = 0.99\ radians](https://tex.z-dn.net/?f=%5Cangle%20QPR%20%3D%2057%5E%5Ccirc%20%5Ctimes%20%5Cdfrac%7B%5Cpi%7D%7B180%7D%20%3D%200.99%5C%20radians)
Using the formula for <em>length of arc</em>:
![l = \theta \times R](https://tex.z-dn.net/?f=l%20%3D%20%5Ctheta%20%5Ctimes%20R)
Where
is the angle subtended by the arc on center.
R is the radius of circle.
Here,
![\theta = 0.99\ radians\\R = 10\ cm](https://tex.z-dn.net/?f=%5Ctheta%20%3D%200.99%5C%20radians%5C%5CR%20%3D%2010%5C%20cm)
![l = 0.99 \times 10\\l = 9.9\ cm](https://tex.z-dn.net/?f=l%20%3D%200.99%20%5Ctimes%2010%5C%5Cl%20%3D%209.9%5C%20cm)
<em>Length of arc QR is </em>
<em> 9.9 cm</em>