CIRCLE
to a circle are equal and
Radius is perpendicular to the tangent
A
B
O
D
C
Q
R
4 cm
4 cm
4 cm
6 cm
8 cm
x
x
A(ΔOBC)
×
1
2
28cm2
=
Sol:
2
=
BC
×
OD
×
1
2
14
×
4
=
=
2
×
1
2
(x + 6)
×
4
2(x + 6)cm2
=
=
2
×
1
2
(x + 8)
×
4
2(x + 8)cm2
=
6 cm
8 cm
A (ΔOAC)
×
1
2
=
AC
×
OQ
A (ΔOAB)
×
1
2
=
AB
×
OR
14 cm
(x + 6)
(x + 8)
D
6 cm
8 cm
Base
height
A(ΔOBC)
A (ΔOAC)
A (ΔOAB)
Q. A triangle ABC is drawn to circumscribe a circle of radius
4cm such that the segments BD and DC into which BC is
divided by the point of contact D are of lengths 8cm and
6cm respectively. Find the sides AB and AC.
Area of triangle = ½ × base × height
Consider ΔOAB
What can you say about BD and BR?
They are tangents from external point B
What can you say about AQ and AR?
Let, AQ = AR = x
They are tangents from external point A
What can you say about CD and CQ?
They are tangents from external point C
Let us find area of ΔOBC
We know, radius is perpendicular to tangent
Consider ΔOAC
Draw OC, OB, and OA
Whenever there is point of contact and centre, we can draw radius
Consider ΔOBC
∴
∴
CD
=
CQ
BD
=
BR
AQ
=
AR
[Tangents from an external point to a circle are equal in length]
Consider point C
We know, tangents from an external point to a circle are equal in length.
Consider point B
=
6 cm
=
8 cm
=
x cm
Consider point A
28
=
+
2x
+
12
+
2x
+
16
56
=
+
4x
A(ΔABC)
4
=
(14 + x)
...(i)
Semi-perimeter of ΔABC (s)
For ΔABC,
=
a + b + c
2
=
x + 6
2
=
(2x + 28)
2
(x + 14) cm
=
A(ΔABC)
A(ΔOBC)
=
+
A(ΔOAC)
+
A(ΔOAB)
28
=
+
2(x + 6)
+
2(x + 8)
A
B
O
D
C
Q
R
4 cm
4 cm
4 cm
6 cm
8 cm
x
x
6 cm
8 cm
14 cm
(x + 6)
(x + 8)
+
14
+
x + 8
Let a = AC = x + 6 ,
b = AB = x + 8,
and c = BC = 14.
=
2 (x + 14)
2
Q. A triangle ABC is drawn to circumscribe a circle of radius
4cm such that the segments BD and DC into which BC is
divided by the point of contact D are of lengths 8cm and
6cm respectively. Find the sides AB and AC.
Sol:
Now, let us consider ΔABC
ΔABC is made up of 3 triangles
ΔOBC,
ΔOAC,
ΔOAB
28cm2
=
A(ΔOBC)
2(x + 6)cm2
=
A(ΔOAC)
=
A(ΔOAB)
2(x + 8)cm2
Now let us apply heron’s
formula to find area of ΔABC