Explanation:
1)∆PQR ~∆PQN. .2)Seg XM || seg QN
To prove that ∆PQR ~ ∆PQN, we need to show that their corresponding sides are proportional in length.
Using the midpoint theorem, we know that PM = MY and PN = NR.
Therefore, we can write:
QR = QP + PR
QP = QR - PR
and
QN = QP/2
PR = QR/2
PN = NR - PR = QR/2 - QR/2 = 0
Using these expressions, we can write:
QP/QR = (QR - PR)/QR = 1 - PR/QR = 1 - 1/2 = 1/2
and
PN/PQ = 0/QP = 0
Since two sides of ∆PQR and ∆PQN are proportional, we can conclude that ∆PQR ~ ∆PQN.
To prove that seg XM || seg QN, we can use the fact that M and N are midpoints of PY and PR, respectively.
Since seg XY || seg QR, we know that ∠QXY = ∠QRP (corresponding angles).
Also, since M is the midpoint of PY, we know that seg PM is parallel to seg QY (midpoint theorem).
Therefore, we have:
∠NQM = ∠QXY (alternate angles)
∠NQM = ∠QRP (corresponding angles)
∠PMX = ∠QYX (alternate angles)
∠PMX = ∠QRP (corresponding angles)
Since two pairs of alternate angles are equal, we can conclude that seg XM || seg QN (by the converse of the alternate interior angles theorem).