Question

Help answer maths questions about circle theorem!

Answer 1

`\sf\\\angle STA=47^o`

Step-by-step explanation:

`\sf\\\bold{Method\ 1:}`

`\sf\\\angle RTA=\angle RQT=73^o\ \ \ [\textsf{Angles in alternate segments are equal.}]\\\\\angle STA=\angle RTA-\angle RTS=73^o-26^o=47^o`

`\bold{Method\ 2:}`

`\sf\\\textsf{Let O be the center of circle. Join O, T and O, R as shown in the figure. }\\\textsf{Now,}\\\angle ROT=2\angle RQT\ \ \ [\textsf{Central angle is twice the inscribed angle on same arc.]}\\\textsf{or, }\angle ROT=2(73^o)=146^o`

`\sf\\\angle OTR=\angle ORT\ \ \ [OT=OR, \textsf{radii of same circle.}]`

`\sf\\\angle ORT+\angle OTR+\angle ROT=180^o\ \ \ [\textsf{Sum of interior angles of triangle is 180}^o.]\\\textsf{or, }\angle OTR+\angle OTR+146^o=180^o\\\textsf{or, }2\angle OTR=34^o\\\textsf{or, }\angle OTR=17^o`

`\sf\\\angle OTA=90^o\ \ \ [\textsf{Radius is perpendicular to tangent at the point of contact.}]`

`\sf\\\textsf{or, }\angle OTR+\angle RTS+\angle STA=90^o\\\textsf{or, }17^o+26^o+\angle STA=90^o\\\textsf{or, }\angle STA=47^o`