Question

In the figure above (not to scale) PT and PS are tangents segments drawn to a circle at T and S respectively TM and SM are chords of the circle If $\angle TMS={100}^{\circ },$ then find the angle between the tangents

• A. ${10}^{\circ }$
• B. ${15}^{\circ }$
• C. ${20}^{\circ }$
• D. ${25}^{\circ }$

Answer 1

Answer: (C)

${20}^{\circ }$

According to the alternate segment theorem, angle between a tangent and a chord through the point of contact is equal to the angle in the alternate segment.

$\angle MSP=\angle MTS,\angle PTM=\angle TSM$

In $△TMS$

$\angle TMS+\angle MST+\angle STM=180$

$⟹\angle MTS+\angle MST=180–100=80$

In $△TSP$

$\angle TSP+\angle SPT+\angle PTS=180$

$\angle TPS=180–(\angle PTM+\angle MTS)–(\angle PSM+\angle MST)$

$\angle TPS=180–2(\angle MTS+\angle MST)=180–2\left(80\right)=20$

Angle between tangents equals ${20}^{\circ }$