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In the Given Figure, Line Pr Touches the Circle at Point Q. Answer the Following Questions with the Help of the Figure. - Geometry Mathematics 2

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प्रश्न

In the given figure, 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.

योग

उत्तर

(1)
Quadrilateral ATSQ is a cyclic quadrilateral.
∴ ∠TAQ + ∠TSQ = 180º            (Opposite angles of a cyclic quadrilateral are supplementary)
(2)
The angle between a tangent of a circle and a chord drawn from the point of contact is congruent to the angle inscribed in the arc opposite to the arc intercepted by that angle.
Here, PR is the tangent and AQ is the chord.
∴ ∠AQP ≅ ∠ATQ and ∠AQP ≅ ∠ASQ
⇒ ∠AQP ≅ ∠ATQ ≅ ∠ASQ
(3)
Angles inscribed in the same arc are congruent.
∴ ∠QTS ≅ ∠SAQ
Also, the angle between a tangent of a circle and a chord drawn from the point of contact is congruent to the angle inscribed in the arc opposite to the arc intercepted by that angle.
Here, PR is the tangent and QS is the chord.
∴ ∠QTS ≅ ∠SQR    
⇒ ∠QTS ≅ ∠SAQ ≅ ∠SQR
(4)
∠TQS = ∠TAS = 65º              (Angles inscribed in the same arc are congruent)
The measure of an inscribed angle is half of the measure of the arc intercepted by it.
∴ ∠TAS = `1/2` m(arc TS)
⇒ m(arc TS) = 2∠TAS = 2 × 65º = 130º
Thus, the measure of ∠TQS is 65º and arc TS is 130º.
(5)
The angle between a tangent of a circle and a chord drawn from the point of contact is congruent to the angle inscribed in the arc opposite to the arc intercepted by that angle.
Here, PR is the tangent and AQ is the chord.
∴ ∠ATQ = ∠AQP = 42º     .
Also, PR is the tangent and QS is the chord.
∴ ∠QTS ≅ ∠SQR = 58º
∠ATQ = ∠ATQ + ∠QTS = 42º + 58º = 100º
Thus, the measure of ∠ATQ is 100º.

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Theorem: Opposite angles of a cyclic quadrilateral are supplementary.
  क्या इस प्रश्न या उत्तर में कोई त्रुटि है?
अध्याय 3: Circle - Problem Set 3 [पृष्ठ ८७]

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बालभारती Geometry (Mathematics 2) [English] 10 Standard SSC Maharashtra State Board
अध्याय 3 Circle
Problem Set 3 | Q 13 | पृष्ठ ८७

संबंधित प्रश्न

Prove that the “the opposite angles of the cyclic quadrilateral are supplementary”.


Prove that “The opposite angles of a cyclic quadrilateral are supplementary”.


In the given figure, ▢PQRS is cyclic. side PQ ≅ side RQ. ∠PSR = 110°, Find -
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(4) measure of ∠PRQ


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MRPN is cyclic, ∠R = (5x – 13)°, ∠N = (4x + 4)°. Find measures of ∠R and ∠N, by completing the following activity.

Solution:

MRPN is cyclic

The opposite angles of a cyclic square are `square`

∠R + ∠N = `square`

∴ (5x – 13)° + (4x + 4)° = `square`

∴ 9x = 189°

∴ x = `square`

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(i) measure of ∠PQR

(ii) m(arc PQR)

(iii) m(arc QR)


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