Answer: [tex]1,719.87\ units^2[/tex]
Step-by-step explanation:
The missing figure is attached.
The formula to calculate the area of trapezoid is:
[tex]A=\frac{h}{2}(B+b)[/tex]
Where "B" is the long base, "b" is the short base and "h" is the height.
1. First we need to find the height DE:
- Since:
[tex]m\angle ADC + m\angle FDA=180\°[/tex]
[tex]m\angle ADC=134\°[/tex]
We can find [tex]m\angle FDA[/tex]. This is:
[tex]m\angle FDA=180\°-134\°=46\°[/tex]
- Observe in the second figure that the triangles EAD and FDA are equal. Then:
[tex]m\angle FDA=m\angle EAD=46\°[/tex]
- Use the Trigonometric Identity [tex]sin\alpha=\frac{opposite}{hypotenuse}[/tex].
In this case:
[tex]\alpha =46\°\\opposite=DE\\hypotenuse=40[/tex]
Then:
[tex]sin(46\°)=\frac{DE}{40}\\\\40*sin(46\°)=DE\\\\DE=28.77\ units[/tex]
2.Use the Trigonometric Identity [tex]cos\alpha=\frac{adjacent}{hypotenuse}[/tex] to find the lenght AE, which is equal to the lenght GB
In this case:
[tex]\alpha =46\°\\adjacent=AE\\hypotenuse=40[/tex]
Then:
[tex]cos(46\°)=\frac{AE}{40}\\\\40*cos(46\°)=AE\\\\AE=GB=27.78\ units[/tex]
- Therefore, the large base AB is:
[tex]AB=AE+DC+GB\\\\AB=27.78+32+27.78\\\\AB=87.56\ units[/tex]
- Now we can substitute values into the formula and calculate the aera of the trapezoid ABCD. This is:
[tex]A=\frac{28.77\ units}{2}(87.56\ units+32\ units)=1,719.87\ units^2[/tex]
