Practice Set 3.4 Circle Class 10th Mathematics Part 2 MHB Solution

Practice Set 3.4

  1. In figure 3.56, in a circle with centre O, length of chord AB is equal to the radius of…
  2. In figure 3.57, is cyclic. side PQ ≅ side RQ. ∠ PSR = 110°, Find-(1) measure of ∠…
  3. is cyclic, ∠R = (5x - 13)°, ∠N = (4x + 4)°. Find measures of angle r and triangle…
  4. In figure 3.58, seg RS is a diameter of the circle with centre O. Point T lies in the…
  5. Prove that, any rectangle is a cyclic quadrilateral.
  6. In figure 3.59, altitudes YZ and XT of∆ WXY intersect at P. Prove that, x^{w}sum t (1)…
  7. In figure 3.60, m(arc NS) = 125°, m(arc EF) = 37°, find the measure ∠ NMS.…
  8. In figure 3.61, chords AC and DE intersect at B. If ∠ ABE = 108°, m(arc AE) = 95°, find…

Practice Set 3.4
Question 1.

In figure 3.56, in a circle with centre O, length of chord AB is equal to the radius of the circle. Find measure of each of the following.



(1) ∠ AOB (2) ∠ ACB

(3) arc AB (4) arc ACB.


Answer:

(1)In ∆AOB,


AB = OA = OB = radius of circle


⇒ ∆AOB is an equilateral triangle


∠ AOB + ∠ ABO + ∠ BAO = 180° {Angle sum property}


⇒ 3∠ AOB = 180° {All the angles are equal}


∠ AOB = 60°


(2)∠ AOB = 2 × ∠ ACB {The measure of an inscribed angle is half the measure of the arc intercepted by it.}


⇒∠ ACB = 30°


(3)∠ AOB = 60°


⇒arc(AB) = 60° {The measure of a minor arc is the measure of its central angle.}


(4) Using Measure of a major arc = 360°- measure of its corresponding minor arc


⇒arc(ACB) = 360° - arc(AB)


⇒arc(ACB) = 360° - 60° = 300°



Question 2.

In figure 3.57,  is cyclic. side PQ ≅ side RQ. ∠ PSR = 110°, Find-



(1) measure of ∠ PQR

(2) m(arc PQR)

(3) m(arc QR)

(4) measure of ∠ PRQ


Answer:

(1) Given PQRS is a cyclic quadrilateral.


∵Opposite angles of a cyclic quadrilateral are supplementary


⇒∠ PSR + ∠ PQR = 180°


⇒∠ PQR = 180° - 110°


⇒∠ PQR = 70°


(2)2 × ∠ PQR = m(arc PR){The measure of an inscribed angle is half the measure of the arc intercepted by it.}


m(arc PR) = 140°


⇒m(arc PQR) = 360° -140° = 220° {Using Measure of a major arc = 360°- measure of its corresponding minor arc}


(3)side PQ ≅ side RQ


∴m(arc PQ) = m(arc RQ){Corresponding arcs of congruent chords of a circle (or congruent circles) are congruent}


⇒m(arc PQR) = m(arc PQ) + m(arc RQ)


⇒m(arc PQR) = 2 × m(arc PQ)


⇒m(arc PQ) = 110°


(4)In ∆ PQR,


∠ PQR + ∠ QRP + ∠ RPQ = 180°{Angle sum property}


⇒∠ PRQ + ∠ RPQ = 180° - ∠ PQR


⇒ 2∠ PRQ = 180° - 70° {∵side PQ ≅ side RQ}


⇒∠ PRQ = 55°



Question 3.

is cyclic, ∠R = (5x - 13)°, ∠N = (4x + 4)°. Find measures of and


Answer:

Given MRPN is a cyclic quadrilateral.


⇒∠ R + ∠ N = 180° {Using Opposite angles of a cyclic quadrilateral are supplementary}


⇒ (5x - 13)° + (4x + 4)° = 180°


⇒9x - 9 = 180°


⇒ x – 1 = 20°


⇒ x = 21°


∠R = (5x - 13)° = 5 × 21 – 13 = 105 – 13 = 92°


∠N = (4x + 4)° = 4 × 21 + 4 = 84 + 4 = 88°



Question 4.

In figure 3.58, seg RS is a diameter of the circle with centre O. Point T lies in the exterior of the circle. Prove that ∠ RTS is an acute angle.




Answer:


Given RS is the diameter


⇒ ∠ ROS = 180°


m(arc RS) = 180°


Now, ∠ RTS is an external angle.





Hence, ∠ RTS is an acute angle.



Question 5.

Prove that, any rectangle is a cyclic quadrilateral.


Answer:


In ABCD,


∠A = 90°{∵ angle of a rectangle is 90°.}


∠C = 90° {opposite angles are equals}


⇒∠ A + ∠ C = 180°


If opposite angles are supplementary, the quadrilateral is cyclic.


∴ ABCD is cyclic.



Question 6.

In figure 3.59, altitudes YZ and XT of

∆ WXY intersect at P. Prove that,



(1) WZPT is cyclic.

(2) Points X, Z, T, Y are concyclic.


Answer:

(1)In WZPT,


∠ WZP = ∠ WTP = 90° {YZ and XT are the altitudes}


If a pair of opposite angles of a quadrilateral is supplementary, then the


quadrilateral is cyclic.


⇒ WZPT is cyclic.


(2)∵ X, Z,T,Y lie on same circle, ∴ they are concyclic.



Question 7.

In figure 3.60, m(arc NS) = 125°, m(arc EF) = 37°, find the measure ∠ NMS.




Answer:

Given m(arc NS) = 125°, m(arc EF) = 37°


Also, ∠ NMS is an external angle, so






Question 8.

In figure 3.61, chords AC and DE intersect at B. If ∠ ABE = 108°, m(arc AE) = 95°, find m(arc DC).




Answer:

Given ∠ ABE = 108°, m(arc AE) = 95°


Using the property of the secant,




⇒m(arc DC) = 108° × 2 – 95°


⇒m(arc DC) = 121°

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