Practice Set 3.4
- In figure 3.56, in a circle with centre O, length of chord AB is equal to the radius of…
- In figure 3.57, is cyclic. side PQ ≅ side RQ. ∠ PSR = 110°, Find-(1) measure of ∠…
- is cyclic, ∠R = (5x - 13)°, ∠N = (4x + 4)°. Find measures of angle r and triangle…
- In figure 3.58, seg RS is a diameter of the circle with centre O. Point T lies in the…
- Prove that, any rectangle is a cyclic quadrilateral.
- In figure 3.59, altitudes YZ and XT of∆ WXY intersect at P. Prove that, x^{w}sum t (1)…
- In figure 3.60, m(arc NS) = 125°, m(arc EF) = 37°, find the measure ∠ NMS.…
- In figure 3.61, chords AC and DE intersect at B. If ∠ ABE = 108°, m(arc AE) = 95°, find…
Practice Set 3.4
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°
