How does one catch up with a rak’ah?

If a person joins the congregation when the imaam is bowing,
one of the following three scenarios applies: 

1-    
He says takbeerat al-ihraam
(saying “Allaahu akbar” to start the prayer) whilst standing, then he bows
whilst the imaam is bowing. In this case he has caught up with the rak’ah
with the imaam.

2-    
He says takbeerat al-ihraam
whilst the imaam is bowing, but he bows after the imaam has stood up from
rukoo’. In this case he is not counted as having caught up with the rak’ah
with the imaam, and he has to make it up.

3-    
He bows straightaway without
saying takbeerat al-ihraam. In this case his prayer is invalid, because he
has omitted one of the pillars or essential parts of the prayer, which is
takbeerat al-ihraam. 

“The fuqaha’ are agreed that whoever catches up with the
imaam in rukoo’ has caught up with the rak’ah, because the Prophet
(peace and blessings of Allaah be upon him) said: ‘Whoever catches up with
rukoo’ has caught up with the rak’ah.’” This was narrated by Abu Dawood, and
classed as saheeh by al-Albaani in Irwa’ al-Ghaleel, 496; on p. 262
he said: one of the things that strengthens the hadeeth is the fact that the
Sahaabah acted in accordance with this: 

1 – Ibn Mas’ood, who said: “Whoever does not catch up with
the imaam when he is bowing has not caught up with that rak’ah.” Its isnaad
is saheeh. 

2 – ‘Abd-Allaah ibn ‘Umar, who said: “If you come when the
imaam is bowing, and you put your hands on you knees before the imaam stands
up, then you have caught up (with that rak’ah).” Its isnaad is saheeh.

3 –  Zayd ibn Thaabit, who used to say: “Whoever catches up
with the rak’ah before the imaam raises his head has indeed caught up with
the rak’ah.” Its isnaad is jayyid. 

See al-Mawsoo’ah al-Fiqhiyyah al-Kuwaitiyyah, 23/133;
al-Mughni, 1/298.