Question

In a triangle ABC (figure) the points P and Q are selected in the sides AC and BC respectively in a way that PC is half of BC and QC is half of AC:`bar(PC)/bar(BC) = 1/2; bar(QC)/bar(AC)= 1/2`. Find `bar(PQ)` if the side `bar(Ab)` is 20?

Answer 1

`PQ=10`

Explanation:

Take `P' = 1/2(C+A)` and `Q'=1/2(C+B)` then

`bar(CP') = bar(CQ)` and `bar(CQ')=bar(CP)` and `abs(bar(P'Q')) = abs(bar(PQ))`
`bar(P'Q')` is parallel to `bar(AB)` and `bar(AB)/(bar(BC))=bar(P'Q')/(bar(CQ'))=bar(PQ)/(bar(CP))`

Finally `bar(PQ)=bar(CP)/(bar(BC)) xx bar(AB) = 20/2 = 10`