SOLUTION: five points O,A,B,C,D are taken in order on a straight line with distances OA=a, OB=b,OC=c,and OD=d. P is a point on the line between B and C and such that AP:PD=BP:PC. Find OP

Algebra ->  Test -> SOLUTION: five points O,A,B,C,D are taken in order on a straight line with distances OA=a, OB=b,OC=c,and OD=d. P is a point on the line between B and C and such that AP:PD=BP:PC. Find OP       Log On


   

Question 1178381: five points O,A,B,C,D are taken in order on a straight line with distances OA=a, OB=b,OC=c,and OD=d.
P is a point on the line between B and C and such that AP:PD=BP:PC.
Find OP in terms of a,b,c,d.
Answer by greenestamps(13200) About Me  (Show Source):
You can put this solution on YOUR website!


Let BP=x. Then

AP = b+x
PD = d-(b+x)
BP = x
PC = c-(b+x)

AP/PD = (b+x)/(d-(b+x))

BP/PC = x/(c-(b+x))

AP/PD = BP/PC:

%28b%2Bx%29%2F%28d-%28b%2Bx%29%29+=+x%2F%28c-%28b%2Bx%29%29

c%28b%2Bx%29-%28b%2Bx%29%5E2+=+dx-bx-x%5E2

cb%2Bcx-b%5E2-2bx-x%5E2+=+dx-bx-x%5E2

cb%2Bcx-b%5E2-2bx+=+dx-bx

cb-b%5E2+=+dx-bx-cx%2B2bx

cb-b%5E2+=+bx%2Bdx-cx

cb-b%5E2+=+x%28b%2Bd-c%29

x+=+%28cb-b%5E2%29%2F%28b%2Bd-c%29



ANSWER: OP+=+bd%2F%28b%2Bd-c%29