Pythagorean theorem (proof from similarity)
The Pythagorean Theorem
The Pythagorean Theorem is the opening topic of Right Triangles & Trigonometry in the California Common Core State Standards. It is aligned to Standard G-SRT.4, which requires students to prove the Pythagorean theorem using triangle similarity.
The Pythagorean theorem states \(a^2+b^2=c^2\) for the legs and hypotenuse of a right triangle, and can be proved from similar triangles.
Theory
In a right triangle, the Pythagorean Theorem relates the two legs \(a,b\) and the hypotenuse \(c\) (the side opposite the right angle):
It can be proved from similarity: the altitude to the hypotenuse splits the triangle into two smaller triangles similar to the whole, and combining their proportions gives the theorem.
The theorem and its rearrangement:
How to use the Pythagorean theorem
- Identify the hypotenuse (opposite the right angle) and the legs.
- Substitute into \(a^2+b^2=c^2\).
- Solve for the unknown side and simplify the radical.
- Converse: to test a right angle, check if the equation holds.
Apply \(a^2+b^2=c^2\).
| \(c^2\) | \(=\) | \(6^2+8^2\) |
| \(=\) | \(36+64=100\) | |
| \(c\) | \(=\) | \(\sqrt{100}=10\) |
Rearrange to solve for the missing leg.
| \(b^2\) | \(=\) | \(13^2-5^2\) |
| \(=\) | \(169-25=144\) | |
| \(b\) | \(=\) | \(\sqrt{144}=12\) |
Check whether the two shorter squares sum to the longest square.
| \(9^2+12^2\) | \(=\) | \(81+144=225\) |
| \(15^2\) | \(=\) | \(225\) |
Equal, so yes — it is a right triangle.
The ladder is the hypotenuse; solve for the height.
| \(h^2\) | \(=\) | \(12^2-5^2=144-25=119\) |
| \(h\) | \(=\) | \(\sqrt{119}\approx 10.9\ \text{ft}\) |
Common pitfalls
Frequently asked questions
What is the Pythagorean theorem?
For a right triangle, \(a^2+b^2=c^2\), where \(c\) is the hypotenuse and \(a,b\) are the legs.
How do you find a leg with the Pythagorean theorem?
Subtract: a leg equals \(\sqrt{c^2-(\text{other leg})^2}\).
What is the converse of the Pythagorean theorem?
If the squares of the two shorter sides sum to the square of the longest side, the triangle is a right triangle.
How is it proved from similarity?
The altitude to the hypotenuse creates two triangles similar to the original; their proportions combine to give \(a^2+b^2=c^2\).