Question

A triangle ABC is drawn to circumscribe a circle of radius 3 cm. such that the segments BD and DC into which BC is divided by the point of contact D are of length 9 cm. and 3 cm. respectively (See adjacent figure). Find the sides AB and AC.

Answer 1

$<b><i><u>$

Construction : Draw radius OB = 3cm
Proof: AC, BC and AB are tangents
BF = BD = 9cm—tangents from B
AF = AB—tangent from A
∴ CD = CB tangents from C
OD = OB = 3cm radius
OBCD forms a square of side 3cm
OD = OB = BC = CD = 3cm
∴ ∠BCD = 90°
BD = 9cm and DC = 3cm
OB = 3cm radius of circle
Let AF = AB = x
Applying Pythagoras
AB = BC + AC
(9 + x)2 = (12)2 + (3 + x)2
81 + 18x + x2 = 144 + 9 + 6x + x2
12x = 72
X = 6cm
∴ AB = 9 + 6 = 15cm
AC = 6 + 3 = 9cm

Answer 2

answer is 6

x=6

6+9=15

6+3=9