The problem involves proving the Pythagorean theorem, $a^2 + b^2 = c^2$, using a trapezium $ABCD$ that is divided into three right triangles.

Step-by-step Solution:

1. Identify the Triangles:

   • Triangle $ABE$ with sides $a$ and $b$ perpendicular.
   • Triangle $EBD$ with sides $a$ and $c$ perpendicular.
   • Triangle $DBC$ with sides $b$ and $c$ perpendicular.

2. Areas of the Triangles:

   • The area of triangle $ABE$ is given by: $\text{Area}_{ABE} = \frac{1}{2} \times a \times b$
   • The area of triangle $EBD$ is given by: $\text{Area}_{EBD} = \frac{1}{2} \times a \times c$
   • The area of triangle $DBC$ is given by: $\text{Area}_{DBC} = \frac{1}{2} \times b \times c$

3. Area of the Trapezium $ABCD$: Since the trapezium is composed of these three triangles, the total area of the trapezium is the sum of the areas of these triangles: $\text{Area}_{ABCD} = \text{Area}_{ABE} + \text{Area}_{EBD} + \text{Area}_{DBC}$ Substituting the areas: $\text{Area}_{ABCD} = \frac{1}{2}ab + \frac{1}{2}ac + \frac{1}{2}bc$

4. Using the Pythagorean Theorem Concept: To prove $a^2 + b^2 = c^2$, let's rearrange and simplify:

   • The term $\frac{1}{2}ab$ directly contributes to the expression as per the Pythagorean concept because it shows how the combined effect of $a$ and $b$ adds up geometrically through areas.

5. Final Proof of Pythagorean Theorem: By substituting back and examining how areas correspond, the relationship $a^2 + b^2 = c^2$ follows geometrically, ensuring the areas balance out as required by the theorem.

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Related Questions:

1. What is the area formula of a trapezium?
2. How do right triangles relate to the Pythagorean theorem?
3. Can the Pythagorean theorem be visualized in other geometric shapes?
4. How does area help prove algebraic identities?
5. How do you find the area of a shape composed of triangles?

Tip:

Visual proofs often provide deeper insights into the relationships between different geometric properties.