Answer:
<u>The area of triangle DEF is equal to 8√3</u>
Step-by-step explanation:
Firstly, side DF is the hypothenuse to the formed right triangle ADE. As side AD has a length of 8 and side AE has a length of 4, we can substitute these values into the Pythagorean theorem:
A^2 + B^2 = C^2
8^2 + 4^2 = C^2
64 + 16 = C^2
80 = C^2
√(80) = √(C^2)
4√5 = C
This means that side DE of triangle DEF is equal to a length of 4√5, and the side DF is also equal to this length because of how squares are congruent as shown in the diagram. This means all we have to do is find the length of side EF.
As side EB and BF is equal to 4, we can create the Pythagorean theorem once again:
A^2 + B^2 = C^2
4^2 + 4^2 = C^2
16 + 16 = C^2
32 = C^2
√(32) = √(C^2)
4√2 = C
Now, we have a triangle to work with! This triangle is triangle DEF and now we have to find the area of the newly defined triangle. To do this, we take the formula for the area of a triangle, A = (HB)/(2), and find values to substitute. Because we do not have height for this triangle, we have to use the Pythagorean theorem once again and hopefully for the last time!
We can bisect side EF with a perpendicular bisector to point D. By doing so, we are creating a line that can be used to solve the height of this triangle. We can now use the theorem:
A^2 + B^2 = C^2
A^2 + (2√2)^2 = (4√2)^2
A^2 + 8 = 32
A^2 = 24
√(A^2) = √24
A = 2√6
Finally! We can now substitute known values into the equation A = (HB)/(2).
A = (HB)/(2)
A = [(2√6)(4√2)]/(2)
A = (16√3)/(2)
A = 8√3
<u>This means that the area of triangle DEF is equal to 8√3!</u>
