Answer:
Step-by-step explanation: Let's height of the parallelogram ABCD as h
height h can be determined by drawing a perpendicular line from one of the vertices of the base.
Now, let's consider triangle ABE and triangle CDF. They have the same base, AB = CD, and the same height, h. Therefore, their areas are equal:
Area(AEB) = Area(CDF) -----------(1)
Since ABCD is a parallelogram, AD || BC. This implies that angle BCD = angle ADB.
angle BCD = angle ADB
angle CDE = angle ADE
CDE is congruent to triangle ADE by the angle-side-angle (ASA) congruence criterion
CE = AE.
given that BF = EF
means that BE = EF + BF = EF + EF = 2EF
Area(CDE) = A - 2x -----------(2)
From equation (1), we have Area(AEB) = Area(CDF), which can be expressed as:
h * AB = h * CD
Since AB = CD, we can cancel the height h, resulting in:
AB = CD
Now, consider the segment DE. We know that DE = AB - AE = AB - CE.
Since AB = CD, we can rewrite DE as:
DE = CD - CE
Substituting the expression for DE in equation (2), we get:
Area(CDE) = A - 2x = Area(CDF) = h * DE = h * (CD - CE)
Substituting the value of CD - CE from the above equation, we have:
A - 2x = h * (CD - CE) = h * DE
Since BF = EF, we know that DE = 2EF.
Substituting this value, we get:
A - 2x = h * 2EF
Dividing both sides by 2, we have:
(A - 2x) / 2 = h * EF
Simplifying further, we get:
(A - 2x) / 2h = EF
But EF = BF, so:
(A - 2x) / 2h = BF
Multiplying both sides by h, we obtain:
(A - 2x) / 2 = h * BF
Substituting the value of h * BF with the area of triangle ABCD (A), we get:
(A - 2x) / 2 = A / 4
Cross-multiplying, we have:
2(A - 2x) = A
Expanding the equation, we get:
2A - 4x = A
Subtracting 2A from both sides, we get:
-4x = -A
Dividing both sides by -4, we have:
x = A / 4
Therefore, the area of triangle EFG (x) is equal to one-fourth the area of parallelogram ABCD (A).