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Standard X

Mathematics

Lengths of Tangents Drawn from External Point

A is a point ...

Question

A is a point at a distance 13 cm from the centre O of a circle of radius 5 cm. AP and AQ are the tangents to the circle at P and Q. If a tangent BC is drawn at a point R lying on the minor PQ to intersect AP at B and AQ at C, find the perimeter of the $Δ A B C$ .

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Solution

BP=BR and CR=CQ ( length of tangents drawn from an external point to a circle are equal )
Perimeter of triangle ABC = AB+BR+RC+CA
=AB+BP+QC+CA
=AP+QA ( AP=QA)
=2AP

In triangle APO, by Pythagoras theorem
$A O^{2} = A P^{2} + P O^{2}$
$13^{2} = A P^{2} + 5^{2}$
$A P^{2} = 144$
$\Rightarrow$ AP = 12

Perimeter of triangle ABC = 2 x 12 = 24cm

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Q. Question 14
A is a point at a distance 13 cm from the centre O of a circle of radius 5 cm. AP and AQ are the tangents to the circle at P and Q . If a tangent BC is drawn at a point R lying on the minor arc PQ to intersect AP at B and AQ at C, find the perimeter of the ∆ABC.

Q. Two tangents are drawn to a circle from an external point

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A is a point at a distance 13 cm from the centre O of a circle of radius 5 cm. AP and AQ are the tangents to the circle at P and Q. If a tangent BC is drawn at a point R lying on the minor PQ to intersect AP at B and AQ at C, find the perimeter of the ΔABC.

BP=BR and CR=CQ ( length of tangents drawn from an external point to a circle are equal ) Perimeter of triangle ABC = AB+BR+RC+CA =AB+BP+QC+CA =AP+QA ( AP=QA) =2AP In triangle APO, by Pythagoras theorem AO2=AP2+PO2 132=AP2+52 AP2=144 ⇒ AP = 12 Perimeter of triangle ABC = 2 x 12 = 24cm

A is a point at a distance 13 cm from the centre O of a circle of radius 5 cm. AP and AQ are the tangents to the circle at P and Q. If a tangent BC is drawn at a point R lying on the minor PQ to intersect AP at B and AQ at C, find the perimeter of the $Δ A B C$ .