七年级学生可以解出这道题吗---**Question Number:** 27
**Question Stem:**
如图，已知在△ABC中，AB=BC，∠ABC=α，(0°＜α＜60°)，射线AM⊥AB，点P为射线AM上的动点(点P不与点A重合)，联结BP，将线段BP绕点B顺时针旋转角度α后，得到线段BQ，联结PQ、QC。
**English Translation of Question Stem:**
As shown in the figure, in △ABC, it is known that AB=BC, ∠ABC=α, (0° ＜ α ＜ 60°), ray AM⊥AB, point P is a moving point on ray AM (point P does not coincide with point A), connect BP, rotate line segment BP clockwise around point B by angle α to obtain line segment BQ, connect PQ and QC.
**Sub-question (1):**
试说明△PAB∽△QCB的理由；
**English Translation of Sub-question (1):**
Try to explain the reason for △PAB∽△QCB;
**Sub-question (2):**
延长QC交射线AM于点D，在点P的移动过程中，∠QDM的大小是否发生变化？若改变请说明理由，若不改变，请求出∠QDM的大小(用含α的代数式表示)；
**English Translation of Sub-question (2):**
Extend QC intersecting ray AM at point D. During the movement of point P, does the size of ∠QDM change? If it changes, please explain the reason. If it does not change, please find the size of ∠QDM (express it using an algebraic expression containing α);
**Sub-question (3):**
当BQ∥AC时，AB=m, AP=n, 过点Q作QE⊥射线AB,垂足为E,那么S△AEQ=______(用含m、n的代数式表示).
**English Translation of Sub-question (3):**
When BQ∥AC, AB=m, AP=n, draw QE⊥ray AB through point Q, with foot of the perpendicular at E, then S△AEQ=______(express it using an algebraic expression containing m and n).
**Geometric Figure Description:**
**Figure 1 (Main Figure):**
* **Type:** Geometric figure/Diagram.
* **Elements:**
* Triangle ABC.
* Ray AM extending upwards from A, perpendicular to AB (indicated by the text AM⊥AB).
* Point P on ray AM.
* Line segments AB, BC, AC, BP, BQ, PQ, QC.
* Point Q.
* Labels: A, B, C, M, P, Q.
* **Relative Position:** Ray AM is drawn vertically upwards from A. AB is drawn horizontally to the right from A. BC is drawn from B, forming angle ABC. C is positioned such that BC is a side of the triangle. P is on AM. Q is positioned after rotating BP around B. Lines PQ and QC are drawn. M is a point on the vertical ray through A.
**Figure 2 (备用图 / Backup Figure):**
* **Type:** Geometric figure/Diagram.
* **Elements:**
* Triangle ABC.
* Ray AM extending upwards from A, perpendicular to AB (indicated by the text AM⊥AB).
* Ray BN extending horizontally to the right from B.
* Labels: A, B, C, M, N.
* **Relative Position:** Ray AM is drawn vertically upwards from A. AB is drawn horizontally to the right from A, extended as ray BN. BC is drawn from B, forming angle ABC. C is positioned such that BC is a side of the triangle. M is a point on the vertical ray through A. N is a point on the horizontal ray extending from B. This figure seems to show the base setup before introducing points P and Q.