Pythagoras’ Inequality Theorems

Key Notes :

If a, b, and c are the side lengths of a triangle and a≤b≤c, then:
a2+b2>c2 if and only if the triangle is acute.
a2+b2=c2 if and only if the triangle is right.
a2+b2<c2 if and only if the triangle is obtuse.

Learn with an example

📚 The sides of a triangle have lengths 6, 8, and 10. What kind of triangle is it?

  • acute
  • right
  • obtuse
  • First, put the three side lengths in order from smallest to largest: a=6, b=8, c=10.
  • To classify the triangle, compare a2+b2 to c2.
  • a2+b2 ? c2
  • 62+82 ? 102
  • 36+64 ? 100
  • 100 = 100
  • Since 62+82=102, these are the side lengths of a right triangle:

📚 The sides of a triangle have lengths 3, 4, and 5. What kind of triangle is it?

  • acute
  • right
  • obtuse
  • First, put the three side lengths in order from smallest to largest: a=2, b=5, c=6.
  • To classify the triangle, compare a2+b2 to c2.
  • a2+b2 ? c2
  • 22+52 ? 62
  • 4+25 ? 36
  • 29 < 36
  • Since 22+52<62, these are the side lengths of an obtuse triangle:

📚 The sides of a triangle have lengths 5, 9, and 9. What kind of triangle is it?

acute

right

obtuse

First, put the three side lengths in order from smallest to largest: a=5, b=9, c=9.
To classify the triangle, compare a2+b2 to c2.
a2+b2 ? c2
52+92 ? 92
25+81 ? 81
106 > 81
Since 52+92>92, these are the side lengths of an acute triangle:

Let’s practice!