Explainer: The Converse of the Pythagorean Theorem

In this explainer, we will learn how to use the converse of the Pythagorean theorem to determine whether a triangle is a right triangle.

You should have already learned about the Pythagorean theorem.

The Pythagorean theorem states that, in a right triangle, the area of a square on the hypotenuse of length 𝑐 is equal to the sum of the areas of the squares on the legs of lengths π‘Ž and 𝑏. Does this mean that a triangle where the lengths of the sides verify the equation π‘Ž+𝑏=π‘οŠ¨οŠ¨οŠ¨ is necessarily a right triangle?

Let us assume that △𝐴𝐡𝐢 is of side lengths π‘Ž, 𝑏, and 𝑐, with π‘Ž+𝑏=π‘οŠ¨οŠ¨οŠ¨. Let △𝐷𝐡𝐢 be a right triangle of side lengths π‘Ž, 𝑏, and 𝑑, with a right angle at 𝐢.

By applying the Pythagorean theorem in △𝐷𝐡𝐢, we find π‘Ž+𝑏=𝑑.

We know that π‘Ž+𝑏=π‘οŠ¨οŠ¨οŠ¨. For the given π‘Ž and 𝑏, π‘Ž+π‘οŠ¨οŠ¨ has only one value. It follows that 𝑐=𝑑.

Since a triangle is uniquely defined by its three sides’ lengths, we conclude that △𝐴𝐡𝐢 and △𝐷𝐡𝐢 are congruent. Consequently, △𝐴𝐡𝐢 has a right angle at 𝐢.

This is a proof of the converse of the Pythagorean theorem.

The Converse of Pythagorean Theorem

If the square of the longest side of a triangle is equal to the sum of the squares of the other two sides, then the triangle is a right triangle.

If π‘Ž+𝑏=π‘οŠ¨οŠ¨οŠ¨, then the triangle has a right angle at 𝐢.

The converse of the Pythagorean theorem allows you to determine whether a given triangle has a right angle or not.

How To Determine Whether a Given Triangle Has a Right Angle or Not

If we are given the three sides of a triangle and want to determine whether the triangle has a right angle or not, the first step is to identify the longest side. Then, we need to check whether the square of the longest side is equal to the sum of the squares of the other two sides or not.

If yes: the triangle has a right angle opposite its longest side.

If not: the triangle has no right angle.

Example 1: Using the Converse of the Pythagorean Theorem

Can the lengths 7.9 cm, 8.1 cm, and 5.3 cm form a right triangle?

Answer

  1. Identify the longest side of the triangle: 8.1 cm.
  2. Square it: 8.1=65.61.
  3. Calculate the sum of the squares of the other two sides: 7.9+5.3=62.41+28.09=90.5.
  4. Compare both numbers: 65.61β‰ 90.5.
  5. Conclude: the given lengths cannot form a right triangle.

Example 2: Using the Converse of the Pythagorean Theorem

Is △𝐴𝐢𝐷 a right triangle at 𝐢?

Answer

  1. Identify the question: does △𝐴𝐢𝐷 have a right angle at 𝐢?
  2. Identify what is given: 𝐴𝐷 and 𝐢𝐷. 𝐴𝐢 is the hypotenuse of △𝐴𝐢𝐷, and 𝐴𝐡 and 𝐡𝐢 are given.
  3. Identify a solving strategy: to answer the question, we will need to apply the converse of the Pythagorean theorem in △𝐴𝐢𝐷, but for this, we need first to find 𝐴𝐢. As it is the hypotenuse of △𝐴𝐡𝐢 and we know the other two sides, we will apply the Pythagorean theorem in △𝐴𝐡𝐢 first.
  4. Implement the solving strategy:
    1. We apply the Pythagorean theorem in △𝐴𝐡𝐢. Let π‘₯ be the length of 𝐴𝐢, the hypotenuse of △𝐴𝐡𝐢. The legs are 17.6 cm and 21 cm. By substituting these into the equation given by the Pythagorean theorem, π‘Ž+𝑏=π‘οŠ¨οŠ¨οŠ¨, we get 17.6+21=π‘₯309.76+441=π‘₯750.76=π‘₯. As π‘₯ is a length, it is positive, and so π‘₯=√750.76=27.4cm.
    2. We apply the converse of the Pythagorean theorem in △𝐴𝐢𝐷.
      1. Identify the longest side of the triangle: 44 cm.
      2. Square it: 44=1,936.
      3. Calculate the sum of the squares of the other two sides: 34+27.4=1,156+750.76=1,906.76.
      4. Compare both numbers: 1,936β‰ 1,906.76.
      5. Conclude: △𝐴𝐢𝐷 has no right angle at 𝐢.

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