过点D作DM⊥x轴于点M, 由题意可得∠BAO = ∠OAF,AO = AF, AB∥OC,则∠BAO= ∠AOF= ∠AFO = ∠OAF,∴∠AOF=60°,∴∠DOM=60°, ∴∠ODM =30°∵OD = AD-OA = AB-OA=6-2=4, ∴MO =2, \( MD=2\sqrt {3} \). ∴D点的坐标为(-2,\( -2\sqrt {3} \)),∴\( k=-2\times (-2\sqrt {3})=4\sqrt {3} \)---The following content is extracted from the image:
**Question Stem:**
8. 如图.四边形 ABCO 是平行四边形, OA = 2, AB = 6, 点 C 在 x 轴的负半轴上, 将 □ABCO 绕点 A 逆时针旋转得到 □ADEF.AD 经过点 O, 点 F 恰好落在 x 轴的正半轴上.若点 D 在反比例函数 y = k/x (x < 0) 的图象上.求 k 的值.
**Mathematical Formulas/Equations:**
y = k/x (x < 0)
**Other Relevant Text:**
(第 8 题)
**Chart/Diagram Description:**
* **Type:** A 2D Cartesian coordinate system.
* **Coordinate Axes:**
* A horizontal X-axis labeled 'x'.
* A vertical Y-axis labeled 'y'.
* The origin is labeled 'O', where the X and Y axes intersect.
* **Points:**
* **A:** A point located on the positive Y-axis.
* **B:** A point located in the second quadrant.
* **C:** A point located on the negative X-axis.
* **D:** A point located in the third quadrant.
* **E:** A point located in the fourth quadrant, visually below point D.
* **F:** A point located on the positive X-axis.
* **M:** A point labeled 'M' located in the second quadrant, between the Y-axis and line segment BC, visually above the X-axis. A dashed line segment connects M to O, and another dashed line segment connects M to a point on the X-axis (which appears to be the projection of M onto the X-axis, though not explicitly labeled).
* **Geometric Shapes:**
* **Parallelogram ABCO:** Formed by vertices A, B, C, and O. Side OA is along the positive Y-axis, and side CO is along the negative X-axis.
* **Parallelogram ADEF:** Formed by vertices A, D, E, and F. This parallelogram is the result of a rotation of parallelogram ABCO around point A.
* **Lines and Segments:**
* Segments OA, AB, BC, CO form parallelogram ABCO.
* Segments AD, DE, EF, FA form parallelogram ADEF.
* The line segment AD is shown passing through the origin O.
* A line segment is drawn from M perpendicular to the X-axis (implied by the dashed lines and the context of typical geometry problems).
* **Curve:**
* A curved line representing a branch of a hyperbola is shown in the third quadrant, passing through point D and extending downwards past point E. This curve corresponds to the function y = k/x for x < 0.
* **Relative Positions:**
* A is above O on the Y-axis.
* C is to the left of O on the X-axis.
* F is to the right of O on the X-axis.
* D is in the third quadrant (x<0, y<0).
* E is in the fourth quadrant (x>0, y<0).