Question

Ajutoor! Multumesc mult!

Answer 1

Explicație pas cu pas:

a)

cel mai simplu se rezolvă prin diferență de arii

BN = CN = 3 cm

BM = AB-AM = 12-8 = 4 cm

$\mathcal{A}_{ABCD} = AB \cdot BC = 12 \cdot 6 = 72 \ cm^{2}$

$\mathcal{A}_{\Delta ADM} = \dfrac{AD \cdot AM}{2} = \dfrac{6 \cdot 8}{2} = \dfrac{48}{2} = 24 \ cm^{2}$

$\mathcal{A}_{\Delta CDN} = \dfrac{CD \cdot CN}{2} = \dfrac{12 \cdot 3}{2} = \dfrac{36}{2} = 18 \ cm^{2}$

$\mathcal{A}_{\Delta MBN} = \dfrac{BM \cdot BN}{2} = \dfrac{4 \cdot 3}{2} = \dfrac{12}{2} = 6 \ cm^{2}$

$\mathcal{A}_{\Delta DNM} = \mathcal{A}_{ABCD} - (\mathcal{A}_{\Delta ADM}+\mathcal{A}_{\Delta CDN}+\mathcal{A}_{\Delta MBN}) = 72 - (24+18+6) = 72 - 48 = \bf 24 \ cm^{2}$

b)

DM² = AD²+AM² = 6²+8² = 36+64 = 100 = 10²

=> DM = 10 cm

DN² = CD²+CN² = 12²+3² = 144+9 = 153

=> DN = 3√17 cm

$\mathcal{A}_{\Delta DNM} = \dfrac{DN \cdot DM \cdot \sin \angle MDN}{2}$

$\dfrac{3 \sqrt{17} \cdot 10 \cdot \sin \angle MDN}{2} = 24 \iff \\ \sin \angle MDN = \dfrac{48}{30 \sqrt{17}} = \bf \dfrac{8 \sqrt{17} }{85}$