CIRCLE
of two tangents drawn from
an external point to a circle are equal.
Q. Prove : Parallelogram circumscribing a circle is a rhombus.
D
A
B
C
P
Q
R
S
To prove : □ABCD is a rhombus.
Hint : prove : AB = BC
Proof :
AR
BR
CP
PD
=
=
=
=
AS
BQ
CQ
DS
…(i)
…(ii)
…(iii)
…(iv)
.
[Length of the tangents drawn from an external point to a circle are equal]
Adding (i), (ii), (iii) & (iv)
AB
+
CD
=
AD
+
BC
(AR + BR)
+ (CP + PD)
=
+ (BQ + CQ)
(AS + SD)
To prove parallelogram a rhombus, adjacent sides should be equal
AB is made up of two segments
AR and
BR
We know, tangent segments drawn from external point are equal
CD is made up of two segments
CP and
DP
We know, tangents drawn from external point are equal
…(v)
[A-R-B, B-Q-C, C-P-D, A-S-D]
∴
AB
+
CD
=
AD
+
BC
Q. Prove : Parallelogram circumscribing a circle is a rhombus.
D
A
B
C
P
Q
R
S
To prove : □ABCD is a rhombus.
Hint : prove : AB = BC
Proof :
…(v)
But,
AB
=
CD
…(vi)
AD
=
BC
…(vii)
[Opposite sides of a
parallelogram are equal]
2AB
=
2BC
AB
=
BC
□ABCD is a rhombus.
∴
∴
AB
+
AB
=
BC
+
BC
In a parallelogram opposite sides are equal
∴
A pair of adjacent sides in a parallelogram is equal
(A parallelogram is a rhombus if a pair of adjacent
sides is equal)
∴