CIRCLE
to a circle are equal and
Radius is perpendicular to the tangent
16 cm
B
P
A
L
O
10 cm
Q. AB is a chord of length 16 cm of a circle of radius 10 cm.
The tangents at A and B intersect at a point P.
Find the length of PA.
Sol.
In ΔAOP
and
ΔBOP
OA
=
OB
(radii of same circle)
OP
=
OP
(common side)
PA
=
PB
[Tangents drawnfrom an external point to a circle are equal length]
∴
ΔAOP
≅
ΔBOP
(SSS test)
∴
∠AOP
=
∠BOP
(c.p.c.t.)
∴
∠AOL
=
∠BOL
…(i)
(P-L-O)
In ΔAOL
and
ΔBOL
OA
=
OB
(radii of same circle)
∠AOL
=
∠BOL
[from (i)]
OL
=
OL
(common side)
?
P
Consider
ΔAOP and ΔBOP
Consider
ΔAOL and ΔBOL
∴
ΔAOL
≅
ΔBOL
(SAS test)
B
P
A
L
O
?
Q. AB is a chord of length 16 cm of a circle of radius 10 cm. The tangents at A and B intersect at a point P.
Find the length of PA.
Sol.
∴
∠OLA
=
∠OLB
(c.p.c.t.)
Let
∠OLA
=
∠OLB
=
x
∠OLA
+
∠OLB
=
1800
(Linear pair)
∴
x
+
x
=
180
∴
2x
=
180
∴
x
=
900
∴
OL
⊥
AB
(Perpendicular from the centre to the
chord, bisects the chord)
∴
AL
=
LB
=
1
2
AB
∴
AL
=
LB
=
8 cm
∴
AL
=
LB
=
1
2
× 16
We know that,
Perpendicular drawn from the centre to the chord, bisects the chord
8
∠OLA and ∠OLB are what type of angle?
Linear pair
AB = 16 cm
AO = 10 cm
x
x