Theory question 103
Circle Theorems(a) In the diagram, TU is a tangent to the circle. RV̂U = 100° and ∠URS = 36°. Calculate the value of angle STU.
(b) In triangle XYZ, |XY| = 5 cm, |YZ| = 8 cm and |XZ| = 6 cm. P is a point on the side XY such that |XP| = 2 cm and the line through P, parallel to YZ meets XZ at Q. Calculate |QZ|.
Show the model answer
Model answer
(a) RVUS is cyclic, so the exterior angle ∠UST = ∠RVU = 100°.
Alternate segment: ∠SUT = ∠URS = 36°.
In ΔSUT: ∠STU = 180° − 100° − 36° = 44°.
(b) PQ ∥ YZ, so XQ/XZ = XP/XY = 2/5. XQ = 2/5 × 6 = 2.4 cm.
|QZ| = 6 − 2.4 = 3.6 cm.
Explanation
The exterior angle of a cyclic quadrilateral equals the interior opposite angle.