How to Solve the Converse of Pythagoras’ Theorem Problems?

The converse of the Pythagoras theorem helps determine whether a triangle is a right triangle or not. While a Pythagorean theorem helps determine the length of the missing side of a right triangle.

Related Topics

• How to Solve Pythagorean Theorem Problems?

A step-by-step guide to the converse of Pythagoras’s theorem

The converse of Pythagoras’ theorem states that if the square of the length of the longest side of a triangle is equal to the sum of the squares of the other two sides, that triangle is right-angled. The converse is the complete reverse of the Pythagorean theorem.

The primary use of the converse of the Pythagorean theorem is that measurements help determine the type of triangle – right, acute, or obtuse. Once the triangle is identified, it becomes easy to make that triangle. Three things happen:

1. If the sum of the squares of the two sides of the triangle is equal to the square of the hypotenuse, the triangle is right-angled.
2. Whether the sum of the squares of the two sides of a triangle is less than the square of the hypotenuse, the triangle is obtuse.
3. If the sum of the squares of the two sides of a triangle is greater than the square of the hypotenuse, the triangle is acute.

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Pythagoras Theorem

Pythagoras’s theorem states that if a triangle is right-angled (\(90\) degrees), the square of the hypotenuse is equal to the sum of the squares of the other two sides. In the given triangle \(ABC\), we have \(BC^2 = AB^2 + AC^2\)​​. Here, ​​​​\(AB\) is the base, \(AC\) is the altitude or the height, and \(BC\) is the hypotenuse. In other words, we can say, in a right triangle, \(\color{blue}{\left(Opposite\right)^2+\:\left(Adjacent\right)^2=\:\left(Hypotenuse\right)^2}\)​​​​​​.

The converse of Pythagoras’ theorem formula

The converse of Pythagoras theorem formula is \(\color{blue}{c^2=a^2 + b^2}\), where \(a, b,\) and \(c\) are the sides of the triangle.

The Converse of Pythagoras’s Theorem – Example 1:

The side of the triangle is of lengths \(8\) units, \(10\) units, and \(6\) units. Is this triangle a right triangle?

Solution:

Using the converse of Pythagoras’ theorem, we obtain,

\((10)^2 = (8)^2 + (6)^2\)

\(100 = 64 + 36\)

\(100=100\)

Since both sides are equal, the triangle is right-angled.

Exercises for the Converse of Pythagoras’s Theorem

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1. The sides of a triangle are \(7, 11\), and \(13\). Check whether the given triangle is a right triangle or not.
2. Determine whether a triangle with sides \(3 cm\), \(5 cm\), and \(7 cm\) is an acute, right or obtuse triangle.
3. Classify a triangle whose side lengths are given as;\(11 in, 13 in\), and \(17 in\).

1. \(\color{blue}{Not}\)
2. \(\color{blue}{Obtuse\:triangle}\)
3. \(\color{blue}{Acute\:triangle}\)