https://file.aiquickdraw.com/mathbot/file/17256270502638bwnuoat.jpg

右の図において、四角形ABCDは正方形である。直線CDについて点Aと反対側に点を、直線CEの上側に2点E，Gを、四角形CEFGが正方形になるようにとる。
これについて、次の問いに答えなさい。
AB=4cm、点Gが辺AD上にあり、CG=5cmであるとき、辺CDを点Dの方に延長した直線上に点pを、PC=CEとなるようにとる。このとき、APDEの面積を求めなさい。 ﻿ 三角形DCE=三角形BCG=2分の1
✕4x4=8（cm）
PC
=CE= 5cmより，CD：DP=4：（5-4）=4：1だから、三角形PDE=4分の1
三角形DCE=2（cm）

が答えの解説だが、これになる理由

In the given diagram, we are tasked with determining the area of quadrilateral \( APDE \). Here's a breakdown of the problem and how we can approach the solution:

### Problem Breakdown:
- \( ABCD \) is a square, with \( AB = 4 \text{ cm} \), so each side of the square is 4 cm.
- Point \( G \) is on side \( AD \), and \( CG = 5 \text{ cm} \).
- Point \( P \) is on the extension of line \( CD \), such that \( PC = CE \).
- The aim is to find the area of quadrilateral \( APDE \).

### Step-by-Step Approach:

1. **Understanding the Square and Extensions:**
   - Since \( ABCD \) is a square, all angles are \( 90^\circ \), and each side is 4 cm.
   - The line \( CG = 5 \text{ cm} \) extends from \( C \), with \( G \) being on the side \( AD \).
   
2. **Use of Triangle Areas:**
   - We are given that \( \triangle DCE = \triangle BCG = \frac{1}{2} \times AB \times AB = \frac{1}{2} \times 4 \times 4 = 8 \text{ cm}^2 \).
   - Also, \( \triangle PDE = \frac{1}{4} \times \triangle DCE \) based on the ratio \( CD : DP = 4:1 \), as \( PC = CE = 5 \text{ cm} \) and \( CD = 4 \text{ cm} \), making the ratio \( 4:1 \).

3. **Calculating the Area of Quadrilateral \( APDE \):**
   - From the above, \( \triangle PDE \) has an area of \( \frac{1}{4} \times 8 = 2 \text{ cm}^2 \).
   - The area of \( \triangle DCE \) is given as 2 cm².
   - Adding these values together gives us the area of quadrilateral \( APDE \).

Thus, the reasoning involves understanding the symmetry of the shapes and their respective areas, and how they relate to each other based on the given conditions.

### Clarifications:
The reason behind this solution is the proportionality between the triangles and the square. By using the given geometric relationships, the areas of smaller triangles within the square and extensions are calculated, leading to the final answer.

### Relative Questions:
1. How do you calculate the area of triangles when the base and height are known?
2. What is the significance of extending a line in geometric problems?
3. How does the proportionality of segments affect the areas of triangles in such diagrams?
4. How do

右の図において、四角形ABCDは正方形である。直線CDについて点Aと反対側に点を、直線CEの上側に2点E，Gを、四角形CEFGが正方形になるようにとる。これについて、次の問いに答えなさい。AB=4cm、点Gが辺AD上にあり、CG=5cmであるとき、辺CDを点Dの方に延長した直線上に点pを、PC=CEとなるようにとる。このとき、APDEの面積を求めなさい。

In this geometric problem, we calculate the area of quadrilateral APDE based on the extension of lines in a square. Given AB = 4 cm, CG = 5 cm, and CD:DP = 4:1, the solution involves using triangle area formulas and proportionality in geometric shapes.

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