Solve the following questions:
In $$\triangle ABC$$ and $$DE\parallel BC$$ and $$AD:DB=2:3$$ then find the ratio of areas of $$\triangle ADE$$ and $$\triangle ABC$$
In $$\triangle ABC$$ $$BC\parallel DE$$ and $$\cfrac{AD}{DB}=\cfrac{2}{3}$$ (given)
In $$\triangle ABC$$ and $$\triangle DEA$$
$$\angle B=\angle D$$
$$\angle C=\angle E$$
AA similarity cirterion
$$\cfrac { ar.\left( \triangle ADE \right) }{ ar.\left( \triangle ABC \right) } =\cfrac { { \left( AD \right) }^{ 2 } }{ { \left( AB \right) }^{ 2 } } =\cfrac { { \left( AD \right) }^{ 2 } }{ { \left( AD+DB \right) }^{ 2 } } $$
$$=\cfrac { { \left( \cfrac { AD }{ BD } \right) }^{ 2 } }{ { \left( \cfrac { AD }{ BD } +1 \right) }^{ 2 } } ={ \left( \cfrac { AD }{ BD } \right) }^{ 2 }\times \cfrac { 1 }{ { \left( \cfrac { AD }{ BD } +1 \right) }^{ 2 } } ={ \left( \cfrac { 2 }{ 3 } \right) }^{ 2 }\times \cfrac { 1 }{ { \left( \cfrac { 2 }{ 3 } +1 \right) }^{ 2 } } =\cfrac { 4 }{ 9 } \times \cfrac { 9 }{ 25 } =\cfrac { 4 }{ 25 } $$