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What Is the Converse of the Pythagorean Theorem?

The converse of the Pythagorean Theorem tells us that by comparing the sum of the squares of two sides of a triangle to the square of its third side, we can determine whether that triangle is an acute, right, or obtuse.

To review, the Pythagorean Theorem is one of the most famous theorems in trigonometry and helps us determine the sides of a right triangle. Some other formulas you might need for triangle relate to finding the base of a triangle and the area. Here are additional articles on how to find the area of a right triangle and one on how to find the third side of a triangle.

Read on to figure out how the converse of the Pythagorean Theorem works!

Pythagorean Theorem, In Reverse

The Pythagorean Theorem tells us that for a right triangle, the sum of the squares of the legs of a triangle is equivalent to the square of the third side:

converse of the pythagorean theorem: right triangle formula

The converse of the Pythagorean Theorem states that if the square of the third side of a triangle is equivalent to the sum of its two shorter sides, then it must be a right triangle. In other words, the converse of the Pythagorean Theorem is the same Pythagorean Theorem but flipped. It gives us an easy way to prove whether a triangle is a right triangle (definition below).

But Wait, There's More!

The converse of Pythagoras' theorem also tells us whether the triangle is acute, obtuse, or right by comparing the sum of the squares of the shorter sides with the square of the longest side of the triangle.

  • If they are equivalent, the triangle is a right triangle. This means the longest side of the triangle lies opposite of the 90 degree or right angle.
  • If the sum of the squares of the shorter sides is larger than the square of the longest side, then the triangle is an acute triangle. The angle formed by the two sides must be less than 90 degrees.
  • If the sum of the squares of the shorter sides is smaller than the square of the longest side, then the triangle is an obtuse triangle. The angle formed by the two sides must be greater than 90 degrees.

By comparing the square of the hypotenuse with the sum of the squares of the two shorter sides, the converse of the Pythagorean Theorem also tells us which type of triangle it is. Let’s take a look at the triangles below.

converse of the pythagorean theorem: types of triangle

Triangle 1 is a right triangle, because:

Right triangle Pythagorean formula

9 + 16 = 25

25 = 25

Triangle 2 is an obtuse triangle, because:

Obtuse triangle Pythagorean formula

16 + 16 < 49

32 < 49

Triangle 3 is an acute triangle, because:

Acute triangle Pythagorean formula

9 + 20.25 > 25

29.25 > 25

The Converse of the Pythagorean Theorem

The converse of the Pythagorean Theorem is the Pythagorean Theorem’s logic applied in reverse. There are three cases that can occur.

  1. If the sum of the squares of two sides of a triangle is equivalent to the square of the hypotenuse, the triangle is a right triangle.

  2. If the sum of the squares of two sides of a triangle is less than the square of the hypotenuse, the triangle is an obtuse triangle.

  3. If the sum of the squares of two sides of a triangle is greater than the square of the hypotenuse, the triangle is an acute triangle.

If you use this reasoning, it’ll be easy to tell what type of triangle you have.

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