Pythagoras' Theorem describes the important relationship between the lengths of the sides of a right-angled triangle.

*Pythagoras' Theorem*

In a right-angled triangle,

*a*^{2} + *b*^{2} = *c*^{2}

The longest side, *c*, in a right-angled triangle is called the *hypotenuse*.

Calculate the length of the side AB of this triangle:

In this triangle,

AB^{2} | = AC^{2} + BC^{2} |

= 5^{2} + 9^{2} | |

= 25 + 81 | |

= 106 |

AB | = = 10.29563014 cm |

= 10.3 cm (to 1 decimal place) |

Calculate the length of the side XY of this triangle.

In this triangle,

YZ^{2} | = XY^{2} + XZ^{2} |

14^{2} | = XY^{2} + 6^{2} |

196 | = XY^{2} + 36 |

XY^{2} | = 160 |

XY | = = 12.64911064 cm |

= 12.6 cm (to 1 decimal place) |

Determine whether or not this triangle contains a right angle.

If the triangle does contain a right angle, then the longest side, BC, would be the hypotenuse.

So, the triangle will be right-angled if AB^{2} + AC^{2} = BC^{2}.

AB^{2} + AC^{2} | = 7^{2} + 14^{2} = 49 + 196 |

= 245 |

BC^{2} | = 19^{2} |

= 361 |

AB^{2} + AC^{2} ≠ BC^{2}