Pythagoras and the right triangle
Use squared lengths to connect the three sides of a right triangle.
Before you begin
- Squares and square roots
- Identifying a right angle
Identify the hypotenuse first
In a right triangle, the side opposite the right angle is the hypotenuse and is the longest side. Pythagoras states that its squared length equals the sum of the squares of the other two sides. Label the hypotenuse before substituting; a rotated diagram does not change which side it is.
Choose addition or subtraction
To find the hypotenuse, add the squares of the two shorter lengths and take a square root. To find a shorter side, subtract the known shorter-side square from the hypotenuse square. The square root converts a squared length back to a length. Check that the result is positive and shorter than the hypotenuse when appropriate.
Worked example
A right triangle has shorter sides 6 cm and 8 cm. Find its hypotenuse.
Show the worked solution
- Square the shorter lengths: 6² = 36 and 8² = 64.
- Add them: c² = 100.
- Take the positive square root because c is a length.
10 cm.
Try it yourself
A right triangle has hypotenuse 13 and one shorter side 5. Find the other side.
Common mistakes
- Use this relationship only for right triangles.
What can you explain now?
Find the diagonal of a rectangle measuring 9 cm by 12 cm.
Compare with the explanation
15 cm.
The diagonal and adjacent sides make a right triangle. Its squared length is 81 + 144 = 225, whose positive square root is 15.
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Common Core mathematics standards ↗