O is the centre of a circle of radius 6 cm. P is a
point such that OP = 10 cm and OP intersects the
circle at T and PC, PD are two tangents drawn to
the circle. If AB is the tangent to the circle at T,
find the length AB.
Answers
Given : O is the centre of a circle of radius 6 cm. P is a point such that OP = 10 cm and OP intersects the circle at T and PC, PD are two tangents drawn to the circle. If AB is the tangent to the circle at T
To Find : the length AB
Solution:
OC = OD = OT = 6 cm = Radius
OP = 10 cm
PT = OP - OT = 10 - 6 = 4 cm
PC is Tangent
Hence OP² = OC² + PC²
=> 10² = 6² + PC²
=> 100 = 36 + PC²
=> 64 = PC²
=> PC = 8
similarly PD = 8 ( PC = PD Equal tangents )
Let say AT = x cm
Then AC = AT = x ( Equal Tangent )
AP = PC - AC
=> AP = 8 - x
AT = c
∠ATP = 90° as ∠OTA = 90°
AP² = AT² + PT²
=> (8 - x)² = x² + 4²
=> 8² + x² - 16x = x² + 16
=> 64 - 16x = 16
=> 4 - x = 1
=> x = 3
AT = 3 cm
Similarly BT = 3 cm
AB = AT + BT = 3 + 3 = 6 cm
AB = 6 cm
length AB = 6 cm
Hope this helps <3
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