辨識這附件中的數學題目
並給我詳細的題目解說
---**Question Stem:**
圖中，ABCD 為一平行四邊形。設 E、F 及 G 分別為 AB、AD 及 CD 上的點使得 BE = 3AE、2DF = 3AF 及 DG = 2CG。已知 BC 的延長線與 EG 的延長線相交於點 H。若 ΔCGH 的面積為 16 cm²，則四邊形 DFEG 的面積為
**Translation of Question Stem:**
In the figure, ABCD is a parallelogram. Let E, F, and G be points on AB, AD, and CD respectively, such that BE = 3AE, 2DF = 3AF, and DG = 2CG. It is known that the extension of BC and the extension of EG intersect at point H. If the area of ΔCGH is 16 cm², then what is the area of quadrilateral DFEG?
**Options:**
A. 46 cm²
B. 49 cm²
C. 105 cm²
D. 115 cm²
**Mathematical Information/Conditions:**
* ABCD is a parallelogram.
* E is on AB, F is on AD, G is on CD.
* BE = 3AE
* 2DF = 3AF
* DG = 2CG
* Line BC is extended to H.
* Line EG is extended to H.
* Area(ΔCGH) = 16 cm²
**Chart/Diagram Description:**
* **Type:** Geometric figure.
* **Main Elements:**
* **Shapes:** A quadrilateral ABCD, which is a parallelogram. Several triangles and another quadrilateral are formed by connecting points.
* **Points:**
* A, B, C, D are the vertices of the parallelogram.
* E is a point on the side AB.
* F is a point on the side AD.
* G is a point on the side CD.
* H is a point outside the parallelogram, formed by the intersection of the extended line BC and the extended line EG.
* **Lines/Segments:**
* The four sides of the parallelogram: AB, BC, CD, DA.
* Segment EF connects point E on AB to point F on AD.
* Segment EG connects point E on AB to point G on CD.
* Segment FC connects point F on AD to point C (implicitly, not drawn as a full segment, but the boundary of quadrilateral DFEG).
* Segment CG is part of CD.
* Segment CH is the extension of BC past C.
* Segment GH is the extension of EG past G.
* Segment EH is the line segment connecting E, G, and H.
* **Relative Positions:**
* Points E, F, G are internal points on their respective sides.
* Point H is to the right and above point C (relative to the orientation shown).
* The line segment EG is extended to meet the extended line segment BC at H, implying E, G, H are collinear.
* The lines AB and CD are parallel.
* The lines AD and BC are parallel.
* The figure shows the arrangement of points and segments as described in the problem statement.