Proving Pythagoras With Graphs and Geometry 

Most students know the formula:

$$a^2 + b^2 = c^2$$

But just memorising Pythagoras’ theorem isn’t enough — seeing why it works makes the idea stick. At Hemel Private Tuition, we prove it in two powerful ways: with geometry and with graphs.

---

🔵 The Geometric Proof

Take four identical right-angled triangles and arrange them inside a square. Depending on how you arrange them, you can form:

• One big square with a smaller square inside, or

• A square split into two smaller squares on the sides.

In both cases, the total area is the same — and the result is the famous equation:

$$a^2+b^2=c^2$$

This visual proof shows students that the theorem is about areas, not just algebra.

---

📊 Proving It with Graphs

We can also use coordinates and graphs. Plot a right-angled triangle on graph paper, for example with points (0,0), (a,0), and (0,b).

• The horizontal distance is a.

• The vertical distance is b.

• The length of the hypotenuse can be found using the distance formula:

$$c = \sqrt{(a-0)^2 + (b-0)^2}$$Squaring both sides gives the same result:$$c^2 = a^2 + b^2$$

This approach ties geometry, algebra, and coordinates together — great practice for GCSE Maths.

---

🎓 Why It Works in Teaching

By combining diagrams, areas, and coordinate geometry, students see that Pythagoras’ theorem isn’t just a “rule to remember,” but something that can be proven in multiple ways. It also builds problem-solving flexibility, a key skill for higher-level maths.