Question

In the figure below PQR is a tangent to the circle at Q. TS is a diameter and TSR and QUV are straight lines. QS is parallel to TV. Angles SQR = 40° and TQV = 55°.

1. Find the following angles giving reasons each case.
   1. <QTS 
   2. <QRS 
   3. <QVT 
   4. <QUT 
2. Given that QR = 8cm, and SR = 4cm. Find the radius of the circle. 

Answer 1

1.  
   1. <QTS = 40°
      Reason: Angles in the alternate segments are equal.
   2. <QRS = 10°
      Reason: Sum of angles in a triangle add up to 180°
   3. <QVT=35°
      Reason: Alternate angles in parallel lines are equal.
   4. <QUT = 50°
      Reason: <QST = 50°, subtended by chord TQ. TQ subtends <QUT, hence <QUT = 50°
2. 8 × 8 = 4(4+x)
   16 + 4x =64
   4x = 48
   x = 12
   Radius = ¹²/₂ = 6cm