Question 1
‘∵∠B′=∠B(Corresonding Line theorem)‘
‘∵∠C′=∠C(Corresponding Line theorem)‘
‘∴△AB′C′∼△ABC( AA ∼)‘
‘∴B′C′AB′=BCAB‘
‘∴1430=2230+x‘
‘14(30+x)=22(30)‘
‘x=1422(30)−30‘
‘x=17.1428571≈17.14‘
‘B′C′AC′=BCAC‘
‘14y=22y+15‘
‘22y=14y+14(15)‘
‘8y=14(15)‘
‘y=26.25‘
Question 2
‘h=bsinA‘
‘h=11.3sin32‘
‘h=5.99‘
‘∵h<6.8<11.3‘
‘∴2△′s exist‘
‘ Lets call point T is the height that is perpendicular on side AB and connects to point C. and B′ be the other possible point of B.‘
‘ Case 1:‘ |
‘∠CB′T=sin−1(6.85.99)‘ |
‘∠CB′T=61.75o‘ |
‘∠AB′C=180−61.75=118.25o( Complentary Angle Theorem)‘ |
‘∠ACB′=180−118.25−32=29.75o(ASTT)‘ |
‘sin∠ACB′AB=sinACB′‘ |
‘sin29.75AB=sin326.8‘ |
‘AB=sin32sin29.75×6.8‘ |
‘AB=6.37‘ |
‘ Case 2:‘
‘∠ABC=61.75‘
‘∠ACB=180−32−61.75=86.25o( ASTT)‘
‘sinCAB=sinACB‘
‘sin86.25AB=sin326.8‘
‘AB=sin32sin86.25×6.8‘
‘AB=12.8‘
Question 3
‘let the square be □ABCD and the inner triangle △AEF‘
‘sin(β)=AEEF=1EF=EF‘
‘sin(α)sin(β)=AEEF×EFEC=AEEC=1EC=EC‘
‘cos(α)sin(β)=EFCF×AEEF=AECF=1CF=CF‘
‘Draw a parallel line to CD that connects point E to AD.sin(α+β)=AECD=1CD=CD‘
‘cos(α)cos(β)=AFAD×AEAF=1AD=AD‘
‘sin(α)cos(β)=AFFD×AEAF=1FD=FD‘
‘cos(β)=AEAF=1AF=AF‘
‘Draw a parallel line to CD that connects point E to AD.cos(α+β)=AEBE=1BE=BE‘