Q13 of 59 Page 83

In figure 3.91, line PR touches the circle at point Q. Answer the following questions with the help of the figure.

(1) What is the sum of ∠ TAQ and∠ TSQ?


(2) Find the angles which are congruent to ∠ AQP.


(3) Which angles are congruent to ∠QTS ?


(4) ∠ TAS = 65°, find the measure of ∠TQS and arc TS.


(5) If ∠AQP = 42°and ∠SQR = 58° find measure of ∠ATS.



(1) As TAQS is a cyclic quadrilateral,


∠TAQ + ∠TSQ = 180° (Sum of opposite angles of a cyclic quadrilateral is 180° )


(2) ∠ASQ and ∠ATQ


(3) ∠ QAS and ∠SQR


(4) ∠TAS = 65°


∠ TQS = ∠ TAS = 65° (angle by same arc TS in the same sector)


m(arc TS) = ∠TQS + ∠TAS


⇒ m(arc TS) = 65 + 65 = 130°


(5) ∠AQP + ∠AQS + ∠SQR = 180°


⇒ 42 + ∠AQS + 58 = 180


⇒ ∠AQS + 100 = 180


⇒ ∠AQS = 80


∠ AQS + ∠ ATS = 180° (opposite angles of a cyclic quadrilateral)


⇒ 80 + ∠ATS = 180


⇒ ∠ATS = 100°


More from this chapter

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11

Draw circles with centres A, B and C each of radius 3 cm, such that each circle touches the other two circles.

 12

Prove that any three points on a circle cannot be collinear.

 14

In figure 3.92, O is the centre of a circle, chord PQ ≅ chord RS If∠ POR = 70°and (arc RS) = 80°, find -

(1) m(arc PR)


(2) m(arc QS)


(3) m(arc QSR)



 15

In figure 3.93, m(arc WY) = 44°,m(arc ZX) = 68°, then

(1) Find the measure of ∠ ZTX.


(2) If WT = 4.8, TX = 8.0,YT = 6.4, find TZ.


(3) If WX = 25, YT = 8,YZ = 26, find WT.