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.