а)
Так как $\Delta AHC - $ прямоугольный, $M - $ середина A C AC A C M H = A C 2 = M C MH=\dfrac{AC}{2}=MC M H = 2 A C = M C
Так как $\Delta BHC - $ прямоугольный, $N - $ середина B C BC B C N H = B C 2 = N C NH=\dfrac{BC}{2}=NC N H = 2 B C = N C
Значит, Δ M C N = Δ M H N \Delta MCN = \Delta MHN Δ M C N = Δ M H N
Тогда
∠ M H N = ∠ M C N = 90 ∘ ⟹ M H ⊥ N H \angle MHN = \angle MCN = 90^\circ \quad \Longrightarrow \quad MH \perp NH ∠ M H N = ∠ M C N = 9 0 ∘ ⟹ M H ⊥ N H
б)
В прямоугольном Δ A B C \Delta ABC Δ A B C C H = A H ⋅ B H = 12 ⋅ 3 = 6 CH = \sqrt{AH\cdot BH} = \sqrt{12\cdot3} = 6 C H = A H ⋅ B H = 12 ⋅ 3 = 6 S Δ A H C = 1 2 ⋅ A H ⋅ H C = 1 2 ⋅ 12 ⋅ 6 = 36 S_{\Delta AHC} = \dfrac{1}{2}\cdot AH\cdot HC = \dfrac{1}{2} \cdot 12 \cdot 6 = 36 S Δ A H C = 2 1 ⋅ A H ⋅ H C = 2 1 ⋅ 12 ⋅ 6 = 36 H M − HM - H M − A B C ABC A B C S Δ M H C = 1 2 ⋅ S Δ A H C = 1 2 ⋅ 36 = 18 S_{\Delta MHC} = \dfrac{1}{2}\cdot S_{\Delta AHC} = \dfrac{1}{2} \cdot 36 = 18 S Δ M H C = 2 1 ⋅ S Δ A H C = 2 1 ⋅ 36 = 18 Δ M H P \Delta MHP Δ M H P cos ∠ P M H = M H M P \cos\angle PMH = \dfrac{MH}{MP} cos ∠ P M H = M P M H Δ C M Q \Delta CMQ Δ C M Q cos ∠ C M H = M C M Q \cos\angle CMH = \dfrac{MC}{MQ} cos ∠ C M H = M Q M C Δ M C H ∼ Δ P Q M по двум пропорциональным сторонам и углу между ними \Delta MCH \sim \Delta PQM \quad \text{по двум пропорциональным сторонам и углу между ними} Δ M C H ∼ Δ P QM по двум пропорциональным сторонам и углу между ними M H M P = M C M Q = k = cos ∠ C M H \dfrac{MH}{MP}=\dfrac{MC}{MQ}=k=\cos \angle CMH M P M H = M Q M C = k = cos ∠ C M H cos ∠ C M H = cos ( 2 ∠ C A H ) = 2 cos 2 ∠ C A H − 1 = 2 A H 2 A H 2 + C H 2 − 1 = 2 ⋅ 12 2 12 2 + 6 2 − 1 = 3 5 \cos \angle CMH = \cos (2\angle CAH) = 2\cos^2 \angle CAH - 1 = \dfrac{2AH^2}{AH^2+CH^2}-1 = \dfrac{2\cdot 12^2}{12^2 + 6^2}-1 = \dfrac{3}{5} cos ∠ C M H = cos ( 2∠ C A H ) = 2 cos 2 ∠ C A H − 1 = A H 2 + C H 2 2 A H 2 − 1 = 1 2 2 + 6 2 2 ⋅ 1 2 2 − 1 = 5 3 S Δ M C H S Δ P Q M = k 2 = cos 2 ∠ C M H = 9 25 \dfrac{S_{\Delta MCH}}{S_{\Delta PQM}} = k^2 = \cos^2 \angle CMH = \dfrac{9}{25} S Δ P QM S Δ M C H = k 2 = cos 2 ∠ C M H = 25 9 S Δ P Q M = 25 9 S Δ M C H = 25 9 ⋅ 18 = 50 S_{\Delta PQM} = \dfrac{25}{9}S_{\Delta MCH} = \dfrac{25}{9}\cdot 18 = 50 S Δ P QM = 9 25 S Δ M C H = 9 25 ⋅ 18 = 50
Ответ: 50 \quad 50 50
