AB is a chord of length 9.6 cm of a circle with radius 6 cm If the tangents at A and B intersect at point P then find the length PA

Asked by Porwal_kumudini | 24th Nov, 2015, 08:58: PM

Expert Answer:

 
 
 
begin mathsize 12px style Solution colon space  Let space the space centre space of space the space circle space be space point space straight O. space Join space OP. space space Length space of space tangents space drawn space from space the space external space point space to space straight a space circle space are space equa space in space length. space Thus comma space AP space equals space BP space Hence comma space triangle space APB space is space an space isosceles space triangle. space PO space is space the space angle space bisector space of space angle APB. space So comma space OP space perpendicular space AB space and space therefore comma space PO space bisects space AB. space Thus comma space AQ space equals space BQ space equals space 9.6 divided by 2 space equals space 4.8 space cm  In space increment AOQ comma By space Pythagoras space theorem comma OQ equals square root of OA squared minus AQ squared end root OQ equals square root of 6 squared minus 4.8 squared end root OQ equals square root of 12.96 end root OQ equals 3.6 space cm  We space know space that space straight a space tangent space is space perpendicular space to space the space radius space at space the space point space of space contact. angle PAQ plus angle OAQ equals 90 degree In space increment APQ comma angle APQ plus angle PAQ equals 90 degree therefore angle PAQ plus angle OAQ equals angle APQ plus angle PAQ therefore angle OAQ equals angle APQ  therefore increment AQP space is space similar space to space increment OQA. therefore AP over OA equals AQ over OQ therefore AP over 6 equals fraction numerator 4.8 over denominator 3.6 end fraction therefore AP equals fraction numerator 4.8 over denominator 3.6 end fraction cross times 6 therefore AP space equals space 8 space cm     end style

Answered by Kajal Gupta | 25th Nov, 2015, 09:49: AM

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