Question

P
А
. In the given figure, PB
PC and QR are tangents
to the circle at B, C and
A respectively. If PB = 8
cm, find the perimeter
(in cm) of the triangle
PQR.
R
B
С​

Answer 1

Answer:

Perimeter of △PQR is 16 cm.

Step-by-step-explanation:

NOTE: Refer to the attachment for the diagram.

In figure,

PB = 8 cm

PB, QR and PC are tangents drawn to a circle.

From Tangent segment theorem, we have,

Tangents PB & PC are drawn from an external point P.

∴ PB = PC - - ( 1 )

Also,

Tangents QA & QB are drawn from an external point Q.

∴ QA = QB - - ( 2 )

Tangents RA & RC are drawn from an external point R.

∴ RA = RC - - ( 3 )

Now, we know that,

Perimeter of triangle = Sum of all sides

∴ P ( △PQR ) = PQ + QR + PR

⇒ P ( △PQR ) = PQ + QA + AR + PR - - [ Q - A - R ]

⇒ P ( △PQR ) = PQ + QB + RC + PR - - [ From ( 2 ) & ( 3 ) ]

⇒ P ( △PQR ) = PB + PC. - - [ P - Q - B, P - R - C ]

⇒ P ( △PQR ) = PB + PB - - [ From ( 1 ) ]

⇒ P ( △PQR ) = 2PB

⇒ P ( △PQR ) = 2 × 8 - - [ Given ]

⇒ P ( △PQR ) = 16 cm

∴ Perimeter of △PQR is 16 cm.

─────────────────────

Additional Information:

1. Tangent theorem:

When a tangent is drawn to a circle, it is always perpendicular to the radius of the circle.

2. Tangent segment theorem:

When two tangent segments are drawn to a circle from an external point, both segments are congruent.

Answer 2

⭐Given:

• PB, PC and QR are targents to the circle at B, C and R respectively.

• PB=8cm

⭐To Find:

• Perimeter of the ∆PQR.

⭐Solution:

PB=PC................ (By tangent therom) ...... (1)

QB=QA............. (By tangent therom) ......... (2)

RA=RC.............. By tangent therom).......... (3)

Since,

We know that,

Perimeter of A triangle=side +side +side

: . Perimeter of ∆ABC=PQ+QR+PR

But QR=QA+QR............ (4)

: . Perimeter of ∆ABC=PQ+QA+QR+PR..... From (2) and (3)

=PQ+QB+RC+PR

=PB+PC

=8+8........ (from 1)

=16

Therefore, Perimeter of a triangle =16 cm