Question

In ΔABC, P is the midpoint of BC, Q divides CA internally in the ratio 2:1 and R divides AB externally in the ratio 1:2, then ______.

Options

• R divides PQ externally in the ratio 2:1

• P, Q, R are collinear

• P divides QR externally in the ratio 3:2

• Q divides PR internally in the ratio 3:2

Answer 1

In ΔABC, P is the midpoint of BC, Q divides CA internally in the ratio 2:1 and R divides AB externally in the ratio 1:2, then P, Q, R are collinear.

Explanation:

`"P"(bar"p")` is midpoint of BC

∴ `bar"p" = (bar"b" + bar"c")/2`

⇒ `2bar"p" = bar"b" + bar"c"`  ......(i)

`"Q"(bar"q")` divides CA internally in the ratio 2:1

∴ `bar"q" = (2bar"a" + bar"c")/3`

⇒ `3bar"q" = 2bar"a" + bar"c"` ......(ii)

`"R"(bar"r")` divides AB externally in the ratio 1:2

`bar"r" = (bar"b" - 2bar"a")/(1 - 2)`

= `(2bar"p" - 3bar"q")/(-1)` ......[From (i) and (ii)]

∴ `bar"r" = -2bar"p" + 3bar"q"`

∴ Points P, Q and R are collinear.