Please solve q 85 ?

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1 Answer
May 20, 2018

9

Explanation:

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For the triangles #Delta PAC# and #Delta PAB# to be similar, all three angles must be the same.

  • It is easy to see that #/_BPA # is common
  • #/_ACP# is obtuse - so it can not be equal to #/_ABP#, and thus it must be equal to the remaining angle, #/_PAB#
  • Finally, we must have #/_PAC=/_ABP#

By the property of similar triangles we have

#AC : AP:PC = AB:BP:AP#

Denoting #PC# and #AP# by #x# and #y#, respectively we have (using #BP = BC+PC = 7+x#)

#6/8 = y/(7+x) = x/y#

Hence #y^2 = x(7+x)# and we have

#6^2/8^2 = x^2/y^2 = x/(7+x)#

and thus

#x/(7+x) = 9/16 qquad implies qquad x/7 = 9/(16-9) = 9/7#

Thus
#x = 9#