In this page we practice understanding the second visual proof of the Pythagorean Theorem by comparing different arrangements of the same right triangles.
We also practice:
- identifying the shapes formed inside the large square
- comparing areas
- understanding why \(a^2 + b^2 = c^2\)
Example 1 – Identify the Shapes
What shapes are formed in the first arrangement?
Answer:
- Two rectangles
- One square with side \(a\)
- One square with side \(b\)
Example 2 – Area in First Shape
Find the total area inside the big square.
\begin{align}\text{Area } = a^2 + b^2\end{align}
Example 3 – Area in Second Shape
Find the area inside the big square.
\begin{align}\text{Area } = c^2\end{align}
Example 4 – Compare Both Shapes
What do you notice about the areas?
Both areas are equal
So:
\begin{align}a^2 + b^2 = c^2\end{align}
Example 5 – Key Idea
Why does the theorem work?
Because we used:
- The same triangles
- The same big square
Only the arrangement changed
Practice
- More Examples (Current page)
Continue Learning
- Pythagorean Formula
- Pythagoras Theorem (a² + b² = c²) – Visual Proof 1
- Pythagoras Theorem (a² + b² = c²) – Visual Proof 2
- Pythagoras Theorem (Example)