How to Solve the Converse of Pythagoras’ Theorem Problems?

The converse of Pythagoras Theorem is the inverse of Pythagoras's theorem and helps distinguish acute, right, or obtuse triangles. In this guide, you can learn the converse of Pythagoras’ theorem.

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.

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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.

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, \(\color{blue}{\left(Opposite\right)^2+\:\left(Adjacent\right)^2=\:\left(Hypotenuse\right)^2}\)​​​​​​.

Proof of converse of Pythagoras theorem

Statement: In a triangle, if the square of one side is equal to the sum of the squares of the other two sides, the angle opposite the first side is right.

Proof: Here, we are given a triangle \(ABC\) in which \(AC^2 = AB^2 + BC^2\). We need to prove that \(∠B = 90°\).

To start with, we construct a \(ΔPQR\) right-angled at \(Q\) such that \(PQ = AB\) and \(QR = BC\).

Now, from \(Δ PQR\), we have:

\(PR^2 = PQ^2 + QR^2\) (Pythagoras Theorem, as \(∠Q=90°)\)

or \(PR^2= AB^2+ BC^2\) (By construction) …………. \((1)\)

But \(AC^2=AB^2+BC^2\) (Given) …………. \((2)\)

So, \(AC = PR\) (From (\(1\)) and (\(2\))) …………. (\(3\))

Now, in \(ΔABC\) and \(ΔPQR\),

\(AB=PQ\) (By construction)

\(BC = QR\) (By construction)

\(AC = PR\) (Proved in (\(3\)))

So, \(ΔABC ≃ ΔPQR\) (According to the \(SSS\) congruence)

\(∠B = ∠Q\) (Corresponding angles of congruent triangles)

\(∠Q = 90°\) (By construction)

So \(∠B = 90°\).

Hence, the opposite of Pythagoras’ theorem is proved.

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

  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 \(17 in\).
This image has an empty alt attribute; its file name is answers.png
  1. \(\color{blue}{Not}\)
  2. \(\color{blue}{Obtuse\:triangle}\)
  3. \(\color{blue}{Acute\:triangle}\)

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