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Question Number 9370 by tawakalitu last updated on 03/Dec/16

Answered by geovane10math last updated on 03/Dec/16

There are infinite triangles with sizes 10 and 15. It depends of inclination of XZ^� M. I think that the value of w depends of XZ^� M. XZ^� M = α Cosine′s Law: w^2 = 15^2 + 10^2 − 2∙10∙15∙cos 𝛂 w^2 = 225 + 100 − 300∙cos 𝛂 w = (√(325 − 300∙cos 𝛂))

$$\mathrm{There}\:\mathrm{are}\:\mathrm{infinite}\:\mathrm{triangles}\:\mathrm{with}\:\mathrm{sizes}\:\mathrm{10} \\ $$$$\mathrm{and}\:\mathrm{15}.\:\mathrm{It}\:\mathrm{depends}\:\mathrm{of}\:\mathrm{inclination}\:\mathrm{of}\:{X}\hat {{Z}M}. \\ $$$$\mathrm{I}\:\mathrm{think}\:\mathrm{that}\:\mathrm{the}\:\mathrm{value}\:\mathrm{of}\:{w}\:\mathrm{depends}\:\mathrm{of}\: \\ $$$${X}\hat {{Z}M}. \\ $$$${X}\hat {{Z}M}\:=\:\alpha \\ $$$$\mathrm{Cosine}'\mathrm{s}\:\mathrm{Law}: \\ $$$$\boldsymbol{{w}}^{\mathrm{2}} \:=\:\mathrm{15}^{\mathrm{2}} \:+\:\mathrm{10}^{\mathrm{2}} \:−\:\mathrm{2}\centerdot\mathrm{10}\centerdot\mathrm{15}\centerdot\mathrm{cos}\:\boldsymbol{\alpha} \\ $$$$\boldsymbol{{w}}^{\mathrm{2}} \:=\:\mathrm{225}\:+\:\mathrm{100}\:−\:\mathrm{300}\centerdot\mathrm{cos}\:\boldsymbol{\alpha} \\ $$$$ \\ $$$$\boldsymbol{{w}}\:=\:\sqrt{\mathrm{325}\:−\:\mathrm{300}\centerdot\mathrm{cos}\:\boldsymbol{\alpha}} \\ $$

Commented by geovane10math last updated on 03/Dec/16

Depends of XM = 8.

$${Depends}\:{of}\:{XM}\:=\:\mathrm{8}. \\ $$

Commented by tawakalitu last updated on 03/Dec/16

I appreciate your effort sir. God bless you.

$$\mathrm{I}\:\mathrm{appreciate}\:\mathrm{your}\:\mathrm{effort}\:\mathrm{sir}.\:\mathrm{God}\:\mathrm{bless}\:\mathrm{you}. \\ $$

Answered by mrW last updated on 03/Dec/16

ΔYXM and ΔYZX are similar, because they have two equal angles. ((∣YX∣)/(∣YZ∣))=((∣XM∣)/(∣ZX∣))=((∣MY∣)/(∣XY∣)) (w/(15))=(8/(10))=((∣MY∣)/w) w=(8/(10))×15=12 cm ∣MY∣=(8/(10))×12=((48)/5)=9.6 cm ∣MZ∣=15−∣MY∣=15−9.6=5.4 cm

$$\Delta\mathrm{YXM}\:\mathrm{and}\:\Delta\mathrm{YZX}\:\mathrm{are}\:\mathrm{similar}, \\ $$$$\mathrm{because}\:\mathrm{they}\:\mathrm{have}\:\mathrm{two}\:\mathrm{equal}\:\mathrm{angles}. \\ $$$$\frac{\mid\mathrm{YX}\mid}{\mid\mathrm{YZ}\mid}=\frac{\mid\mathrm{XM}\mid}{\mid\mathrm{ZX}\mid}=\frac{\mid\mathrm{MY}\mid}{\mid\mathrm{XY}\mid} \\ $$$$\frac{\mathrm{w}}{\mathrm{15}}=\frac{\mathrm{8}}{\mathrm{10}}=\frac{\mid\mathrm{MY}\mid}{\mathrm{w}} \\ $$$$\mathrm{w}=\frac{\mathrm{8}}{\mathrm{10}}×\mathrm{15}=\mathrm{12}\:\mathrm{cm} \\ $$$$\mid\mathrm{MY}\mid=\frac{\mathrm{8}}{\mathrm{10}}×\mathrm{12}=\frac{\mathrm{48}}{\mathrm{5}}=\mathrm{9}.\mathrm{6}\:\mathrm{cm} \\ $$$$\mid\mathrm{MZ}\mid=\mathrm{15}−\mid\mathrm{MY}\mid=\mathrm{15}−\mathrm{9}.\mathrm{6}=\mathrm{5}.\mathrm{4}\:\mathrm{cm} \\ $$

Commented by tawakalitu last updated on 03/Dec/16

Thanks you so much sir. God bless you.

$$\mathrm{Thanks}\:\mathrm{you}\:\mathrm{so}\:\mathrm{much}\:\mathrm{sir}.\:\mathrm{God}\:\mathrm{bless}\:\mathrm{you}. \\ $$

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