First read the pesukim in Zecharia which describe the Four Fast Days (not including Yom Kippur and Taanit Esther):
(19) ’Thus saith the Lord of hosts: The fast of the fourth month, and the fast of the fifth, and the fast of the seventh, and the fast of the tenth, shall be to the house of Judah joy and gladness, and cheerful seasons; therefore love ye truth and peace.
Which months are referred to by the navi Zecharia? Which might refer to Tzom Gedalia?
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Now let's read the story in Tanach, which describes the death of Gedalia, governer of Judea after the First Beit HaMikdash had already been destroyed:
According to the pesukim above, when was Gedalia killed? Why do you think his murder was important enough for us to establish a fast day?
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Next read the following Gemara:
צום השביעי זה ג' בתשרי שבו נהרג גדליה בן אחיקם ומי הרגו ישמעאל בן נתניה הרגו ללמדך ששקולה מיתתן של צדיקים כשריפת בית אלהינו
“The fast of the seventh,” this is the third of Tishrei, on which Gedaliah, son of Ahikam, was killed. And who killed him? Ishmael, son of Nethaniah, killed him. The Sages established a fast to commemorate Gedaliah’s death to teach you that the death of the righteous is equivalent to the burning of the Temple of our Lord.
From the above Gemara, on what exact day does it sound like Gedalia was killed?
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Now read the classic commentary Radak:
רד"ק ירמיהו פרק מא
(א) ויהי בחדש השביעי... ובר"ה נהרג גדליה בן אחיקם וקבעו התענית במוצאי ר"ה מפני שהוא י"ט:
According to Radak, on what exact day does it sound like Gedalia was killed? Why do you think this makes Tzom Gedalia so significant?
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