Pythagorean theorem (proof from similarity)
The Pythagorean Theorem
The Pythagorean Theorem is the opening topic of Right Triangles & Trigonometry in the Texas Essential Knowledge and Skills (Geometry, §111.41). It is aligned to Standard G.6(D), which requires students to prove the Pythagorean theorem and its converse and apply them to solve problems.
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 \(a^2+b^2=c^2\).