Question

18. In figure 3.96 P is the point of contact.
(1) If m(arc PR) = 140°,
POR = 36°,
find m(arc PQ)
(2) If OP = 7.2, OQ = 3.2,
find OR and QR
(3) If OP = 7.2, OR = 16.2,
find QR.​

Answer 1

(1)Join PQ.

The measure of an inscribed angle is half of the measure of the arc intercepted by it.

∴ ∠PQR = 1/2 m(arc PR)

= 1/2×140°

= 70º

In ∆POQ,

∠PQR = ∠POQ + ∠OPQ --------------- ( Remote interior angle theorem )

⇒ 70º = 36º + ∠OPQ

⇒ ∠OPQ = 70º − 36º = 34º

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.

∴ ∠PRQ = ∠OPQ = 34º

Now,

∠PRQ = 1/2 m(arc PQ) --------------- (The measure of an inscribed angle is half of the measure of the arc intercepted by it)

⇒ m(arc PQ) = 2∠PRQ = 2 × 34º = 68º

$\boxed{m(arc PQ) = 68°}$

(2)

OP is the tangent and OQR is the secant.

∴ OQ × OR = OP2 --------------- (Tangent secant segment theorem)

⇒ 3.2 × OR = (7.2)2

⇒ OR = 7.2×7.2 / 3.2 = 16.2

$\boxed{OR = 16.2}$

∴ QR = OR − OQ = 16.2 − 3.2 = 13

(3)

OP is the tangent and OQR is the secant.

∴ OQ × OR = OP2 --------------- (Tangent secant segment theorem)

⇒ OQ × 16.2 = (7.2)2

⇒ OQ = 7.2×7.2 / 16.2 = 3.2

∴ QR = OR − OQ = 16.2 − 3.2 = 13

$\boxed{QR = 13}$

Answer 2

Answer:

Mark as brainlyest

Step-by-step explanation:

(1) The measure of an inscribed angle is half of the measure of the arc intercepted by it.

∴∠PQR=

2

1

m(arcPR)=

2

1

×140

o

=70

o

In △POQ,

∠PQR=∠POQ+∠OPQ (Measure of an exterior angle of a triangle is equal to the sum of its remote interior angles)

⇒70

o

=36

o

+∠OPQ∠OPQ

⇒70

o

−36

o

=34

o

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.

∴∠PRQ=∠OPQ=34

o

Now,

∠PRQ=

2

1

m(arcPQ) (The measure of an inscribed angle is half of the measure of the arc intercepted by it)

⇒m(arcPQ)=2∠PRQ=2×34

o

=68

o

(2) OP is the tangent and OQR is the secant.

∴OQ×OR=OP

2

⇒3.2×OR=(7.2)

2

⇒OR=

3.2

7.2×7.2

(Tangent secant segment theorem)

⇒QR=OR−OQ=162−32=13

(3) OP is the tangent and OQR is the secant. ... OQ×OR=OP

2

⇒OQ×16.2=(7.2)

2

OQ=

16.2

7.2×7.2

=3.2(Tangent secant segment theorem)

∴QR=OR−OQ=16.2−32=13