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+++ title = "Pythagoras' theorem" description = "Use Pythagoras' theorem and its converse in 2D problems and coordinate settings." template = "chapter.html" weight = 23 [extra] subject = "math" chapter_id = "pythagoras-theorem" strand = "Measures, Shape & Space" +++
Pythagoras' theorem links the side lengths of a right-angled triangle. It is one of the most useful tools in geometry and coordinate problems.
Pythagoras' theorem
In a right triangle with legs \(a,b\) and hypotenuse \(c\):
\[ a^2+b^2=c^2. \]
The hypotenuse is always the side opposite the right angle and is the longest side.
Finding a missing side
Example 1. If \(a=6\), \(b=8\), then:
\[ c=\sqrt{6^2+8^2}=\sqrt{36+64}=\sqrt{100}=10. \]
Example 2. If \(c=13\), \(a=5\), then:
\[ b=\sqrt{13^2-5^2}=\sqrt{169-25}=\sqrt{144}=12. \]
Converse of Pythagoras
If side lengths satisfy \(a^2+b^2=c^2\), the triangle is right-angled.
Example: \(7,24,25\) gives \(7^2+24^2=49+576=625=25^2\), so right-angled.
Classifying triangle type
For largest side \(c\):
- \(a^2+b^2=c^2\): right triangle.
- \(a^2+b^2>c^2\): acute triangle.
- \(a^2+b^2<c^2\): obtuse triangle.
Using in coordinate geometry
Distance between \(A(x_1,y_1)\) and \(B(x_2,y_2)\):
\[ AB=\sqrt{(x_2-x_1)^2+(y_2-y_1)^2}. \]
This is Pythagoras applied to horizontal and vertical changes.
Typical applications
- Diagonal of rectangles/squares.
- Ladders against walls.
- Shortest straight-line distances on maps/grids.
- Checking whether corners are right angles in construction.
Exam strategy
- Identify the right angle first.
- Label hypotenuse clearly before substitution.
- Keep exact surd form unless decimal is requested.
- Check answer size: hypotenuse must be longest.
History
The theorem is named after Pythagoras, but knowledge of right-triangle relationships existed in earlier Babylonian and Indian mathematics. It became central in Euclidean geometry and later in coordinate methods.
Derivation and reasoning
Area-based proof idea
A classic proof rearranges four identical right triangles inside a square in two different ways. Equating uncovered areas leads to \(a^2+b^2=c^2\).
Checkpoints
Find hypotenuse if legs are 9 and 12.
Answer: \(c=\sqrt{9^2+12^2}=\sqrt{225}=15\).
A right triangle has hypotenuse 17 and one leg 8. Find the other leg.
Answer: \(\sqrt{17^2-8^2}=\sqrt{225}=15\).
Are sides 10, 24, 26 right-angled?
Answer: Yes, \(10^2+24^2=100+576=676=26^2\).
Distance between \((-2,1)\) and \((4,9)\).
Answer: \(\sqrt{6^2+8^2}=10\).
Rectangle has side lengths 7 and 24. Find diagonal.
Answer: \(\sqrt{7^2+24^2}=25\).
Applications
- Architecture: checking right angles and structural diagonals.
- Navigation: shortest displacement calculations on grid maps.
- Computer graphics: pixel distance and vector magnitude.
References
- Standard high-school geometry curricula on Pythagoras and its converse.
- Classical Euclidean geometry references.