Hi Friends,
Need help in solving the below problem, I have just started solving the problems so it would be of great help if you could give explanations also.
In the figure given,P is a point on AB, such that AP:PB = 4:3 PQ is parallel to AC and QD is parallel to CP.What is ratio AP:PD ???
Similar Triangles problem.
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Initially, consider one pair of 2 similar triangles ACP and PQD
Then consider another pair of 2 similar triangles PCB and DQB
Required ratio AP/PD is equal to PB/BD. We know that PB is 3, then PB/BD=3/x provided we set BD as x. AP=4 and AP/PD=4:y provided we set y as PD.
PB=3=x+y, PD=y=3-x, thus, 3/x=4/(3-x).
Solve for x and then find y for the required proportion of 4/y
3/x=4/(3-x), 3(3-x)=4x, 9-3x=4x, 7x=9, x=9/7 and y=3-9/7=12/7
Now proportion 4/y=4 : 12/7= 4* 7/12=7/3 is the answer
Then consider another pair of 2 similar triangles PCB and DQB
Required ratio AP/PD is equal to PB/BD. We know that PB is 3, then PB/BD=3/x provided we set BD as x. AP=4 and AP/PD=4:y provided we set y as PD.
PB=3=x+y, PD=y=3-x, thus, 3/x=4/(3-x).
Solve for x and then find y for the required proportion of 4/y
3/x=4/(3-x), 3(3-x)=4x, 9-3x=4x, 7x=9, x=9/7 and y=3-9/7=12/7
Now proportion 4/y=4 : 12/7= 4* 7/12=7/3 is the answer
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I also came up with 7:3.
Looked at it like this:
Given: AP:PB = 4:3 and the parallel line info, then we know PD:DB=4:3 also.
So PD = 4/7*PB. Call AP 4 & PB 3. Then PD = 12/7
So AP:PD = 4:12/7 = 28/7:12/7 = 28:12 = 7:3
Looked at it like this:
Given: AP:PB = 4:3 and the parallel line info, then we know PD:DB=4:3 also.
So PD = 4/7*PB. Call AP 4 & PB 3. Then PD = 12/7
So AP:PD = 4:12/7 = 28/7:12/7 = 28:12 = 7:3
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Since AC is parallel to PQ, ∆ABC is similar to ∆PBQ.
Since AB:PB = 7:3, AC:PQ = 7:3.
Since CP is parallel to QD, ∆ACP is similar to ∆PQD.
Since AC:PQ = 7:3, AP:PD = 7:3.
Since AC is parallel to PQ, ∆ABC is similar to ∆PBQ.
Since AB:PB = 7:3, AC:PQ = 7:3.
Since CP is parallel to QD, ∆ACP is similar to ∆PQD.
Since AC:PQ = 7:3, AP:PD = 7:3.
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Using properties of similar triangles,
AP/PB = CQ/QB
Also, CQ/QB = PD/DB=4/3
PD/DB=4/3 =>DB/PD = 3/4=> DB/PD+1 = 7/4=> (DB+PD)/PD = 7/4 = > PB/PD = 7/4
PD = (4/7)PB AND AP = (4/3)PB
AP/PD = 7/3
AP/PB = CQ/QB
Also, CQ/QB = PD/DB=4/3
PD/DB=4/3 =>DB/PD = 3/4=> DB/PD+1 = 7/4=> (DB+PD)/PD = 7/4 = > PB/PD = 7/4
PD = (4/7)PB AND AP = (4/3)PB
AP/PD = 7/3
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