In a trapezium PQRS, PQ || SR and S=R. Prove that (i) PS = QR, (ii) PR = QS.
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Answer:
Construct a line to join diagonal QS
Construct a line to join diagonal QSDiagonal QS intersect the line MN at point O
Construct a line to join diagonal QSDiagonal QS intersect the line MN at point OIt is given that PQ∥SR and MN∥PQ
Construct a line to join diagonal QSDiagonal QS intersect the line MN at point OIt is given that PQ∥SR and MN∥PQWe can write it as
Construct a line to join diagonal QSDiagonal QS intersect the line MN at point OIt is given that PQ∥SR and MN∥PQWe can write it asPQ∥MN∥SR
Construct a line to join diagonal QSDiagonal QS intersect the line MN at point OIt is given that PQ∥SR and MN∥PQWe can write it asPQ∥MN∥SRConsider △SPQ
Construct a line to join diagonal QSDiagonal QS intersect the line MN at point OIt is given that PQ∥SR and MN∥PQWe can write it asPQ∥MN∥SRConsider △SPQWe know that MO∥PQ and M is the midpoint to the side SP
Construct a line to join diagonal QSDiagonal QS intersect the line MN at point OIt is given that PQ∥SR and MN∥PQWe can write it asPQ∥MN∥SRConsider △SPQWe know that MO∥PQ and M is the midpoint to the side SPO is the midpoint of the line QS
Construct a line to join diagonal QSDiagonal QS intersect the line MN at point OIt is given that PQ∥SR and MN∥PQWe can write it asPQ∥MN∥SRConsider △SPQWe know that MO∥PQ and M is the midpoint to the side SPO is the midpoint of the line QSWe know that MN∥SR
Construct a line to join diagonal QSDiagonal QS intersect the line MN at point OIt is given that PQ∥SR and MN∥PQWe can write it asPQ∥MN∥SRConsider △SPQWe know that MO∥PQ and M is the midpoint to the side SPO is the midpoint of the line QSWe know that MN∥SRIn △QRS we know that ON∥SR
Construct a line to join diagonal QSDiagonal QS intersect the line MN at point OIt is given that PQ∥SR and MN∥PQWe can write it asPQ∥MN∥SRConsider △SPQWe know that MO∥PQ and M is the midpoint to the side SPO is the midpoint of the line QSWe know that MN∥SRIn △QRS we know that ON∥SRO is the midpoint of the diagonal QS
Construct a line to join diagonal QSDiagonal QS intersect the line MN at point OIt is given that PQ∥SR and MN∥PQWe can write it asPQ∥MN∥SRConsider △SPQWe know that MO∥PQ and M is the midpoint to the side SPO is the midpoint of the line QSWe know that MN∥SRIn △QRS we know that ON∥SRO is the midpoint of the diagonal QSHence, based on the converse mid-point theorem we know that N is the midpoint of QR
Construct a line to join diagonal QSDiagonal QS intersect the line MN at point OIt is given that PQ∥SR and MN∥PQWe can write it asPQ∥MN∥SRConsider △SPQWe know that MO∥PQ and M is the midpoint to the side SPO is the midpoint of the line QSWe know that MN∥SRIn △QRS we know that ON∥SRO is the midpoint of the diagonal QSHence, based on the converse mid-point theorem we know that N is the midpoint of QRtherefore it is proved that N is the midpoint of QR
