帮我讲解图片上面的数学题---**Question 23. (Total 13 points for this question)**
**Question Stem:**
As shown in the figure, in a Cartesian coordinate system, given the area of triangle AOB, S△AOB = 8, OA = OB, BC = 7, and the coordinates of point P are (a, 6).
**Sub-questions:**
(1) Find the coordinates of the three vertices of △ABC: A (0,4) B (-4,0) C (3,0).
(2) When point P is on the extension of CA, find the value of AP/AC.
(3) Point P is in the first quadrant. Connect BP intersecting AC at point Q. If the area of △PAQ is equal to the area of △QBC, find the coordinates of point P.
**Other Relevant Text/Calculations/Annotations:**
**(For Part 2):**
S△COP = S△AOP + S△AOC
3 * 6 * (1/2) = 4 * PH * (1/2) + 4 * 3 * (1/2)
PH = 3/2
∴ P (-3/2, 6)
∴ AP/AC = 1/2 (Consideration: Junior high students can do segment ratio)
**(For Part 3):**
S△PAQ = S△QBC
∴ S△PAQ + S△ABQ = S△QBC + S△ABQ
S△PAB = S△ABC = 14
Using the same method, find H: (2,6)
∴ PH = |a-2|
When P is to the right of H:
S△PAB = (a-2) * 6 * (1/2) - (a-2) * 2 * (1/2) = 2a - 4 = 14
2a - 4 = 14
2a = 18
a = 9
When P is to the right of H, move to here (referring to the calculation for 'a').
∴ P (9, 6)
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**Chart/Diagram Description:**
**Chart 1 (Top Diagram, related to Question 2):**
* **Type:** Geometric figure in a 2D Cartesian coordinate system.
* **Main Elements:**
* **Coordinate Axes:** A horizontal X-axis and a vertical Y-axis intersect at the origin O.
* **Points:**
* O: The origin (0,0).
* A: A point on the positive Y-axis.
* B: A point on the negative X-axis.
* C: A point on the positive X-axis.
* P: A point located in the second quadrant, with its Y-coordinate labeled '6'.
* H: A point on the horizontal line passing through P, located on the Y-axis.
* **Lines:**
* Solid line segments form triangle ABC.
* A horizontal dashed line passes through P and H, parallel to the X-axis, indicating the Y-coordinate '6'.
* A dashed line extends from P, passes through A, and continues downwards. This line represents the extension of CA.
* **Labels and Annotations:** 'x' and 'y' labels for the axes. The number '6' is marked on the Y-axis, indicating the Y-coordinate of P and H.
**Chart 2 (Bottom Diagram, related to Question 3):**
* **Type:** Geometric figure in a 2D Cartesian coordinate system.
* **Main Elements:**
* **Coordinate Axes:** A horizontal X-axis and a vertical Y-axis intersect at the origin O.
* **Points:**
* O: The origin (0,0).
* A: A point on the positive Y-axis.
* B: A point on the negative X-axis.
* C: A point on the positive X-axis.
* P: A point located in the first quadrant, with its Y-coordinate labeled '6'.
* H: A point on the horizontal line passing through P, with its X-coordinate labeled '2'. It is annotated with "23题" (Question 23).
* Q: The intersection point of line segment BP and line segment AC.
* **Lines:**
* Solid line segments form triangle ABC.
* Solid line segment BP connects B to P.
* Solid line segment AC connects A to C.
* A horizontal dashed line passes through P and H, parallel to the X-axis, indicating the Y-coordinate '6'.
* Solid line segments AQ and CQ are formed by the intersection Q.
* Solid line segment PQ.
* A dashed line from H to A is indicated, forming a triangle with base on the horizontal line at y=6.
* **Labels and Annotations:** 'x' and 'y' labels for the axes. The number '6' is marked on the Y-axis, indicating the Y-coordinate of P and H. The number '2' is marked on the X-axis (implicitly, as the X-coordinate of H). "23题" is next to H.