Question

Given the right trapezoid calculate angle `theta` and the area of triangle `hat(EAD)`, provided `EA=4, AB=BC=CD=DA=2, AB_|_EC`?

Answer 1

`/_theta = arctan((sqrt(3)-1)/2)~~0.350879411 rad`

`S_(EAD) = 2`

Explanation:

From right triangle `Delta ABE`, knowing hypotenuse `AE=4` and cathetus `AB=2` we can find another cathetus:
`BE=sqrt(4^2-2^2)=sqrt(12)=2sqrt(3)`.

In the right triangle `Delta DCE` cathetus `CD=2`. Second cathetus `CE = CB+BE = 2+2sqrt(3)`

Now we can determine tangent of angle `/_ theta`:
`tan(theta) = (CD)/(CE) = 2/(2+2sqrt(3)) = 1/(sqrt(3)+1)=(sqrt(3)-1)/2`
Angle `/_theta` can be determined using an inverse function `tan^(-1)()` or, as it is often expressed, `arctan()`:
`/_theta = arctan((sqrt(3)-1)/2)~~0.350879411 rad`

Area of triangle `Delta EAD` can be calculated using the length of its base `AD` and altitude `DC`:

`S_(EAD) = 1/2*AD*DC = 1/2*2*2 = 2`