Two sides AB and BC and median AM triangle ABC are respectively equal to sides PQ and QR and median PN of APQR Show that: (i) AABM = APQN (ii) AABC = APQR
Answers
△ABC and△PQR in which AB=PQ,BC=QR and AM=PN.
Since AM and PN are median of triangles ABC and PQR respectively.
Now, BC=QR ∣ Given
⇒ 21
BC= 21
QR ∣ Median divides opposite sides in two equal parts
BM=QN... (1)
Now, in △ABM and△PQN we have
AB=PQ ∣ Given
BM=QN ∣ From (i)
and AM=PN ∣ Given
∴ By SSS criterion of congruence, we have
△ABM≅△PQN, which proves (i)
∠B=∠Q ... (2) ∣ Since, corresponding parts of the congruent triangle are equal
Now, in △ABC and△PQR we have
AB=PQ ∣ Given
∠B=∠Q ∣ From (2)
BC=QR ∣ Given
∴ by SAS criterion of congruence, we have
△ABC≅△PQR, which proves (ii)
Explanation:
(i) In ∆ABM and ∆PQN,
AB = PQ ...(1) | Given
AM = PN ...(2) | Given
BC = QR
⇒ 2BM = 2QN
| ∵ M and N are the mid-points of BC and QR respectively
⇒ BM = QN ...(3)
In view of (1), (2) and (3),
∆ABM ≅ ∆PQN | SSS Rule
(ii) ∵ ∆ABM ≅ ∆PQN
| Proved in (1) above
∴ ∠ABM = ∠PQN | C.P.C.T.
⇒ ∠ABC = ∠PQR ...(4)
In ∆ABC and ∆PQR,
AB = PQ | Given
BC = QR | Given
∠ABC = ∠PQR | From (4)
∴ ∆ABC ≅ ∆PQR. | SAS Rule