Question Number 132131 by Ar Brandon last updated on 11/Feb/21
Commented by Ar Brandon last updated on 11/Feb/21
$$\mathrm{Find}\:\mathrm{the}\:\mathrm{relation}\:\mathrm{between}\:\mathrm{the}\:\mathrm{areas}\:\mathrm{of}\: \\ $$$$\mathrm{triangle}\:\mathrm{ABC}\:\mathrm{and}\:\mathrm{triangle}\:\mathrm{A}'\mathrm{B}'\mathrm{C}' \\ $$
Commented by mr W last updated on 11/Feb/21
$${if}\:{A},{B},{C}\:{are}\:{the}\:{midpoints},\:{then} \\ $$$$\frac{\Delta_{{ABC}} }{\Delta_{{A}'{B}'{C}'} }=\frac{\mathrm{1}}{\mathrm{7}} \\ $$
Commented by Ar Brandon last updated on 11/Feb/21
$$\mathrm{Sir},\:\mathrm{can}\:\mathrm{you}\:\mathrm{prove}\:\mathrm{the}\:\mathrm{theory}\:? \\ $$$$\mathrm{I}\:\mathrm{tried},\:\mathrm{but}\:\mathrm{got}\:\mathrm{stucked}. \\ $$
Commented by mr W last updated on 11/Feb/21
Commented by mr W last updated on 11/Feb/21
$${each}\:{of}\:{the}\:\mathrm{7}\:{small}\:{triangles}\:{has}\:{the} \\ $$$${same}\:{area}. \\ $$
Commented by mr W last updated on 11/Feb/21
$${example}: \\ $$$${A}'{B}={BA}\:\Rightarrow\Delta_{{A}'{BC}} =\Delta_{{ABC}} \\ $$$${B}'{C}={CB}\:\Rightarrow\Delta_{{A}'{B}'{C}} =\Delta_{{A}'{BC}} =\Delta_{{ABC}} \\ $$
Commented by Ar Brandon last updated on 11/Feb/21
$$\mathrm{OK}\:\mathrm{Sir} \\ $$
Commented by Ar Brandon last updated on 11/Feb/21
Thanks