KnowledgeBoat Logo

Mathematics

In the figure (ii) given below, two concentric circles with centre O are of radii 5 cm and 3 cm. From an external point P, tangents PA and PB are drawn to these circles. If AP = 12 cm, find BP.

In the figure (ii) given below, two concentric circles with centre O are of radii 5 cm and 3 cm. From an external point P, tangents PA and PB are drawn to these circles. If AP = 12 cm, find BP. Circles, ML Aggarwal Understanding Mathematics Solutions ICSE Class 10.

Circles

7 Likes

Answer

Join OA and OB.

In the figure (ii) given below, two concentric circles with centre O are of radii 5 cm and 3 cm. From an external point P, tangents PA and PB are drawn to these circles. If AP = 12 cm, find BP. Circles, ML Aggarwal Understanding Mathematics Solutions ICSE Class 10.

OA ⊥ AP. (∵ tangent at a point and radius through the point are perpendicular to each other.)

So, in right angled triangle OAP,

OP2=OA2+AP2OP2=52+122OP2=25+144OP2=169OP=169OP=13 cm.OP^2 = OA^2 + AP^2 \\[1em] OP^2 = 5^2 + 12^2 \\[1em] OP^2 = 25 + 144 \\[1em] OP^2 = 169 \\[1em] OP = \sqrt{169} \\[1em] OP = 13 \text{ cm}.

OB ⊥ BP. (∵ tangent at a point and radius through the point are perpendicular to each other.)

So, in right angled triangle OBP,

OP2=BP2+OB2132=BP2+32169=9+BP2BP2=160BP=160BP=410 cm.OP^2 = BP^2 + OB^2 \\[1em] 13^2 = BP^2 + 3^2 \\[1em] 169 = 9 + BP^2 \\[1em] BP^2 = 160 \\[1em] BP = \sqrt{160} \\[1em] BP = 4\sqrt{10} \text{ cm}.

Hence, the length of BP = 4104\sqrt{10} cm.

Answered By

4 Likes


Related Questions