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Lengths of Tangents Drawn from External Point

In the given ...

Question

$In the given figure, a circle touches the side BC of Δ A B C at P. AB and AC are two tangents drawn from a point A outside the circle. If AQ = 15 cm, find the perimeter of Δ A B C .$

60 cm

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40 cm

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30 cm

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35 cm

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Solution

The correct option is C 30 cm
From the figure,
$A Q = A R$
[Tangents drawn from a point outside the circle are equal in length]

$\Rightarrow A Q = A C + C R (∵ A R = A C + C R)$

$C R = C P$
[Tangents drawn from a point outside the circle are equal in length]

$∴ A Q = A C + C P . . . (i)$

$Also, A Q = A B + B Q$

$But, B Q = B P$
[Tangents drawn from a point outside the circle are equal in length]

$∴ A Q = A B + B P . . . (i i)$

$Adding equations (i) and (ii),$

$2 A Q = A C + C P + B P + A B$

$2 A Q = Perimeter of Δ A B C$

$∴ Perimeter of Δ A B C = 2 \times 15 = 30 c m$

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In the given figure, a circle touches the side BC of ΔABC at P. AB and AC are two tangents drawn from a point A outside the circle. If AQ = 15 cm, find the perimeter of ΔABC.

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