Pre-Algebra
Geometry
The Pythagorean theorem
20 practice questions
0 video lessons
Theory + worked examples
The Pythagorean Theorem
Texas Pre-Algebra (TEKS) • Standard 8.7(C) • Geometry
The Pythagorean Theorem is a topic in Geometry in the Texas Essential Knowledge and Skills. It is aligned to Standard 8.7(C), which requires students to apply the Pythagorean theorem to find side lengths of right triangles.
In a right triangle, the squares of the legs add to the square of the hypotenuse: \(a^2+b^2=c^2\).
Theory
The Pythagorean theorem relates the sides of a right triangle: \(a^2+b^2=c^2\), where \(c\) is the hypotenuse.
The hypotenuse is the longest side, opposite the right angle.
A 3-4-5 right triangle.
The Pythagorean theorem.
The theorem:
\[a^2+b^2=c^2\]
Only for right triangles.
How to use the theorem
- Identify the legs and hypotenuse.
- Square the known sides.
- Add for the hypotenuse, subtract for a leg.
- Take the square root.
Example 1 — Find the hypotenuse
Legs \(3\) and \(4\). Find \(c\).
Solution
\(a^2+b^2=c^2\).
| \(9+16\) | \(=\) | \(25\) |
| \(c\) | \(=\) | \(5\) |
Example 2 — Find a leg
Hypotenuse \(13\), leg \(5\). Find the other leg.
Solution
Subtract squares.
| \(13^2-5^2\) | \(=\) | \(144\) |
| \(b\) | \(=\) | \(12\) |
Example 3 — Check right
Is a \(6,8,10\) triangle right?
Solution
\(36+64=100=10^2\) — yes.
Example 4 — Estimate
Legs \(1\) and \(1\). Find \(c\).
Solution
\(c=\sqrt2\approx1.41\).
| \(c\) | \(=\) | \(\sqrt2\) |
Common pitfalls
\(c\) is the hypotenuse, always the longest side.
Only works for right triangles.
Take the square root at the end.
Frequently asked questions
What is the Pythagorean theorem?
\(a^2+b^2=c^2\) for a right triangle.
What is the hypotenuse?
The longest side, opposite the right angle.
Does it work for any triangle?
No, only right triangles.
Legs \(3\) and \(4\): hypotenuse?
\(5\).
← Previous subtopic
Scale drawings and geometric figures
Next subtopic →
Distance between two points (Pythagorean theorem application)
More in Geometry