Question

In ΔABC, X is the mid-point of AB, and Y is the mid-point of AC. BY and CX are produced and meet the straight line through A parallel to BC at P and Q respectively. Prove AP = AQ.

Answer 1

Join X and Y
In ΔABP,
X and Y are the mid-points of AB and AC respectively
Therefore, XY || BC
Since BC || AP
⇒ XY || AP and XY || AQ

∴ XY = `(1)/(2)"AP"` .......(i)

XY = `(1)/(2)"AQ"` ...........(ii)

From (i) and (ii)
⇒ `(1)/(2)"AP" = (1)/(2)"AQ"`

⇒ AP = AQ.