Question

coplanar squares abgh and bcdf are adjacent, with cd=10 units and ah=5 units. point e is on segments ad and gb what is the area of triangle abe, in square units? express your answer in a common fraction.

Answer 1

`\boxed{\boxed{\text{Area}_(ABE)=(25)/(3)\ unit^2}}`

Solution-

ΔABE ~ ΔACD by AA (Angle-Angle) similarity, as

1. m∠BAE = m∠CAD (as ∠A is common to both)
2. m∠ABE = m∠ACD = 90° (As each angles of a square is 90°)

According to the similarity of triangles,

`\Rightarrow (AB^2)/(AC^2)=\frac{\text{Area}_(ABE)}{\text{Area}_(ACD)}`

`\Rightarrow (AB^2)/((AC)^2)=\frac{\text{Area}_(ABE)}{(1)/(2)* AC* CD}`

`\Rightarrow (AB^2)/((AB+BC)^2)=\frac{\text{Area}_(ABE)}{(1)/(2)* (AB+BC)* CD}`

`\Rightarrow (5^2)/((5+10)^2)=\frac{\text{Area}_(ABE)}{(1)/(2)* (5+10)* 10}`

`\Rightarrow (5^2)/(15^2)=\frac{\text{Area}_(ABE)}{(1)/(2)* 15* 10}`

`\Rightarrow (25)/(225)=\frac{\text{Area}_(ABE)}{75}`

`\Rightarrow \text{Area}_(ABE)=(25* 75)/(225)`

`\Rightarrow \text{Area}_(ABE)=(25)/(3)\ unit^2`