12 MN, ZMNK, ZKMN,\nZMTN-?

Ответ: MN = 9√3, ∠MNK = 30°, ∠KMN = 30°, ∠MTN = 90°

Разбираемся:

1. В \(\triangle\)MKE \(\angle\)MKE = 90°, \(\angle\)MEK = 120°, значит \(\angle\)KME = 180° - 90° - (180° - 120°) = 30°.
2. ME = MK\(\cdot\)cos30° = 9\(\cdot\)(\(\sqrt{3}\)/2) = 4.5\(\sqrt{3}\).
3. MN = 2ME = 9\(\sqrt{3}\).
4. В \(\triangle\)MNK MK = MN, значит \(\triangle\)MNK равнобедренный, \(\angle\)KMN = \(\angle\)KNM = (180° - 120°)/2 = 30°.
5. Тогда \(\angle\)MNK = 30°, а \(\angle\)MTN = 180° - 30° - 60° = 90°.

Ответ: MN = 9√3, ∠MNK = 30°, ∠KMN = 30°, ∠MTN = 90°