3. Two sides AB and BC and median AM of one triangle ABC are respectively equal to sides PQ and QR and median PN of \small \Delta PQR (see Fig). Show that:
              (i)  \small \Delta ABM \cong \Delta PQN
              (ii) \small \Delta ABC \cong \Delta PQR

   

Answers (1)

(i)  From the figure we can say that :

                                             BC\ =\ QR

or                                          \frac{1}{2}BC\ =\ \frac{1}{2}QR

or                                            BM\ =\ QN

Now, consider \Delta ABM   and   \Delta PQN,

(a)  AM\ =\ PN                        (Given)

(b)   AB\ =\ PQ                          (Given)

(c)   BM\ =\ QN                        (Prove above)

Thus by SSS congruence rule, we can conclude that :

                                                           \small \Delta ABM \cong \Delta PQN

(ii)   Consider \Delta ABC  and   \Delta PQR :

(a)  AB\ =\ PQ                  (Given)

(b)   \angle ABC\ =\ \angle PQR   (by c.p.c.t. from the above proof)

(c)   BC\ =\ QR                 (Given)

Thus by SAS congruence rule,

                                                \small \Delta ABC \cong \Delta PQR

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