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ответ: а) прямые СН⊥ CF - доказано. б) LM =2√2 (ед. длины)
Объяснение:
Треугольники АСN и МСВ - прямоугольные и равнобедренные по построению.
В ⊿ АСВ катет ВС=4, катет АС=8
В ⊿ МСN катет МС=4, катет CN=8
ВС=МС, АС=NC;⇒⊿ АСВ =⊿ МСN по 1-му признаку, их сходные острые углы равны.
а) В прямоугольном треугольнике высота, проведенная из прямого угла к гипотенузе, делит его на два подобных друг другу и исходному.
⊿ FCM≈⊿ АСВ≈⊿ АСН ⇒ их сходные углы равны.
Сумма острых углов прямоугольного треугольника равна 90°⇒
Угол FCM+угол АСН=90°, что и требовалось доказать.
б) В ⊿ АLM сторона АМ=АС-МС=8-4=4; углы при АМ равны по 45°, т.к. ∠АМL=∠CMB - вертикальные, ∠МАL =45° как угол равнобедренного ⊿АСN⇒
⊿ АLM - равнобедренный, ∠АLM=90°.⇒
Катет LM=АМ•sin45°=4•√2/2=2√2 (ед. длины)
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