In the adjoining figure, PQRS is a trapezium in which PQ || SR and M is the midpoint of PS. A line segment MN || PQ meets QR at N.
Question:
In the adjoining figure, PQRS is a trapezium in which PQ || SR and M is the midpoint of PS. A line segment MN || PQ meets QR at N. Show that N is the midpoint of QR.
Solution:
Given: In trapezium PQRS, PQ || SR, M is the midpoint of PS and MN || PQ.
To prove: N is the midpoint of QR.
Construction: Join QS.
Proof:
In ∆SPQ,
Since, M is the mid-point of SP and MO || PQ.
Therefore, O is the mid-point of SQ. (By Mid-point theorem)
Similarly, in ∆SRQ,
Since, O is the mid-point of SQ and ON || SR (SR || PQ and MN || PQ)
Therefore, N is the mid-point of QR. (By Mid-point theorem)