The Converse of the Pythagorean Theorem

In this lesson, we will learn

• the converse of the Pythagorean Theorem
• how to use the converse to determine whether a triangle is acute, right or obtuse
• how to prove the converse of the Pythagorean Theorem

The Converse of the Pythagorean Theorem

The Pythagorean Theorem states that

In any right triangle, the sum of the squared lengths of the two legs is equal to the squared length of the hypotenuse.

The converse of the Pythagorean Theorem states that

For any triangle with sides a, b, c, if a² + b² = c², then the angle between a and b measures 90° and the triangle is a right triangle.

How to use the converse to determine the type of triangle

We can also use the converse of the Pythagorean theorem to check whether a given triangle is an acute triangle, a right triangle or an obtuse 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 triangle, i.e. the angle facing side c is an acute angle.

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

If c² > a² + b² then it is an obtuse 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, right or obtuse 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 triangle.

Example :

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

Example :

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

The Converse of the Pythagorean Theorem
This video discusses the converse of the Pythagorean Theorem and how to use it verify if a triangle is a right triangle. Also, two triangle inequalities used to classify a triangle by the lengths of its sides. Both are related to the Pythagorean Theorem.

How to Use the Converse of the Pythagorean Theorem
How to determine if three given lengths of the sides of a triangle make a right triangle. This is an application of the Converse of the Pythagorean Theorem.

Converse of Pythagorean Theorem

Proof of the Converse of the Pythagorean Theorem

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