The Converse of the Pythagorean Theorem

We can also use the Pythagorean theorem to check whether a given triangle is an acute-angled triangle, a right-angled triangle or an obtuse-angled triangle.

For a triangle with sides a, b and c and c is the longest side then:

If c² < a² + b² then it is an acute-angled triangle, i.e. the angle facing side c is an acute angle.

If c² = a² + b² then it is a right-angled triangle, i.e. the angle facing side c is a right angle.

If c² > a² + b² then it is an obtuse-angled triangle, i.e. the angle facing side c is an obtuse angle.

Example :

Determine whether a triangle with sides 3 cm, 5 cm and 7 cm is an acute-angled, right-angled or obtuse-angled triangle.

Solution:

We choose the two shorter sides to be a and b and the longest side to be c.

So a = 3, b = 5 and c = 7.

a² + b² = 3² + 5² = 9 + 25 = 34

c² = 7² = 49

49 > 34 → c² > a² + b², and so the triangle is an obtuse-angled triangle.

Example :

Determine whether a triangle with sides 12 cm, 14 cm and 18 cm is an acute-angled, right-angled or obtuse-angled triangle.

Solution:

We choose the two shorter sides to be a and b and the longest side to be c.

So a = 12, b = 14 and c = 18.

a² + b² = 12² + 14² = 144 + 196 = 340

c² = 18² = 324

340 < 34 → c² < a² + b², and so the triangle is an acute-angled triangle.

Example :

Determine whether a triangle with sides 8 cm, 15 cm and 17 cm is an acute-angled, right-angled or obtuse-angled triangle.

Solution:

We choose the two shorter sides to be a and b and the longest side to be c.

So a = 8, b = 15 and c = 17.

a² + b² = 8² + 15² = 64 + 225 = 289

c² = 17² = 289

289 = 289 → c² = a² + b², and so the triangle is an right triangle.

Videos

Solving with the pythagorean theorem -
Professor Edward Burger explains solving with the pythagorean theorem

The pythagorean theorem - Professor Edward Burger explains antoher example with the pythagorean theorem

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