Advertisement Remove all ads

ABCD is a parallelogram. E, F are the midpoints of BC and CD respectively. AE, AF meet the diagonal BD at Q and P respectively. Show that P and Q trisect DB. - Mathematics and Statistics

Advertisement Remove all ads
Advertisement Remove all ads
Sum

ABCD is a parallelogram. E, F are the midpoints of BC and CD respectively. AE, AF meet the diagonal BD at Q and P respectively. Show that P and Q trisect DB.

Advertisement Remove all ads

Solution

Let A, B, C, D, E, F, P, Q have position vectors `bar"a",bar"b",bar"c",bar"d",bar"e",bar"f",bar"p",bar"q"` respectively.

∵ ABCD is a parallelogram

∴ `bar"AB" = bar"DC"`

∴ `bar"b" - bar"a" = bar"c" - bar"d"`

∴ `bar"c" = bar"b" + bar"d" - bar"a"`        ....(1)

E is the midpoint of BC

∴ `bar"e" = (bar"b" + bar"c")/2`

∴ `2bar"e" = bar"b" + bar"c"`     ....(2)

F is the mid-point of CD

∴ `bar"f" = (bar"c" + bar"d")/2`

∴ `2bar"f" = bar"c" + bar"d"`        .....(3)

`2bar"e" = bar"b" + bar"c"`      ...[By (2)]

`= bar"b" + (bar"b" + bar"d" + bar"a")`    ....[By (1)]

∴ `2bar"e" + bar"a" = 2bar"b" + bar"d"`

∴ `(2bar"e" + bar"a")/(2+1) = (2bar"b" + bar"d")/(2 + 1)`

LHS is the position vector of the point on AE and RHS is the position vector of the point on DB. But AE and DB meet at Q.

∴ `bar"q" = (2bar"b" + bar"d")/(2 + 1)`

∴ Q divides DB in the ratio 2 : 1      ....(4)

`2bar"f" = bar"c" + bar"d"`      ....[By(3)]

`= (bar"b" + bar"d" - bar"a") + bar"d"`      ....[By(1)]

∴ `2bar"f" + bar"a" = 2bar"d" + bar"b"`

∴ `(bar"a" + 2bar"f")/(1 + 2) = (bar"b" + 2bar"d")/(1 + 2)`

LHS is the position vector of the point on AF and RHS is the position vector of the point on DB. But AF and DB meet at P.

∴ `bar"p" = (bar"b" + 2bar"d")/(1 + 2)`

∴ P divides DB in the ratio 1 : 2       .....(5)

From (4) and (5), if follows that P and Q trisect DB.

Concept: Vectors and Their Types
  Is there an error in this question or solution?
Advertisement Remove all ads

APPEARS IN

Advertisement Remove all ads
Share
Notifications

View all notifications


      Forgot password?
View in app×