To solve the problem, let's analyze the given image and the task step by step.
Step 1: Understand the Geometry
The image shows three squares:
1. A blue square with side length \( a \).
2. A red square with side length \( b \).
3. A green square with side length \( c \).
The vertices of these squares are connected in such a way that:
- The blue square is positioned at the bottom left.
- The red square is positioned above the blue square, sharing one vertex with it.
- The green square is positioned to the right of the blue square, sharing one vertex with it.
The angle \( \angle ACB \) is given as \( 90^\circ \), which means that the line segment \( AB \) forms a right angle with the line segment \( CB \).
Step 2: Identify Key Points
- Let the side length of the blue square be \( a \).
- Let the side length of the red square be \( b \).
- Let the side length of the green square be \( c \).
From the image:
- Point \( A \) is a vertex of the blue square.
- Point \( B \) is a vertex of the red square.
- Point \( C \) is a vertex of the blue square and also a vertex of the green square.
Step 3: Use the Pythagorean Theorem
Since \( \angle ACB = 90^\circ \), the triangle \( \triangle ACB \) is a right triangle. We can use the Pythagorean theorem to relate the sides of this triangle.
- The length of \( AC \) is the side length of the blue square, which is \( a \).
- The length of \( CB \) is the side length of the green square, which is \( c \).
- The length of \( AB \) is the hypotenuse of the right triangle \( \triangle ACB \).
Using the Pythagorean theorem:
\[
AB^2 = AC^2 + CB^2
\]
Step 4: Relate \( AB \) to the Red Square
The length \( AB \) is also the side length of the red square, which is \( b \). Therefore:
\[
b^2 = a^2 + c^2
\]
Step 5: Conclusion
The relationship between the areas of the squares is given by the Pythagorean theorem. The area of the red square is equal to the sum of the areas of the blue square and the green square.
Thus, the solution is:
\[
\boxed{b^2 = a^2 + c^2}
\]
Parent Tip: Review the logic above to help your child master the concept of pythagorean theorem with squares.
