Converse of Pythagoras theorem

In this chapter we will learn the converse of Pythagoras theorem with solved examples.

Let us first review the basics of Pythagoras theorem.

Pythagoras theorem review

Pythagoras theorem states that for right angled triangle the following formula is applicable;

$\mathtt{hypotenuse^{2} =\ ( side\ 1) \ ^{2} +\ ( side\ 2)^{2}}$

The formula states that the square of hypotenuse of right triangle is equal to sum of square of other two sides.

For example, see the below right angle triangle ABC.

According to Pythagoras theorem;

$\mathtt{AC^{2} =\ AB^{2} \ +\ BC^{2}}$

The converse states that if square of longest side of triangle is equal to sum of square of other two sides then the triangle is a right triangle.

In other words, if triangle sides satisfy below expression, then the given triangle is a right triangle.

$\mathtt{( Side\ 3)^{2} =\ ( side\ 1)^{2} \ +\ ( side\ 2)^{2} \ }$

where;
Side 3 = longest side of triangle

Proof of Pythagoras theorem converse

Given is the triangle ABC such that; $\mathtt{AC^{2} =\ AB^{2} \ +\ BC^{2}}$

Prove that triangle ABC is right angled triangle (i.e. ∠ABC = 90 degree)

Solution
Draw triangle PQR such that;
PQ = AB
QR = BC
∠PQR = 90

Taking triangle PQR.
Since triangle PQR is right angled triangle we will apply Pythagoras theorem.

$\mathtt{PR^{2} =PQ^{2} +QR^{2}}$ – eq (1)

It’s already given that;
$\mathtt{AC^{2} =\ AB^{2} \ +\ BC^{2}}$

As, AB = PQ and BC = QR

we can write;
$\mathtt{AC^{2} =\ PQ^{2} \ +\ QR^{2}}$ – eq(2)

Comparing equation (i) and (ii), we get;
$\mathtt{PR^{2} =AC^{2}}\\\ \\ \mathtt{PR\ =\ AC}$

Now taking triangle ABC and PQR for congruency;
AB = PQ
BC = QR
PR = AC

Hence by SSS congruency, both triangles are congruent.

We know that in congruent triangles, all sides and angles are equal.

So, ∠Q = ∠B = 90 degree.

Hence, if any triangle follows Pythagoras formula then it will be a right triangle.

Pythagoras theorem converse – Solved Problems

(01) Check if the below triangle measures 90 degree.

Solution
The above triangle may look like 90 degree triangle but don’t be fast in getting into conclusion.

The ∠Q may measure 89 or 91 degree. This slight difference in angle may not qualify it as right triangle.

Best way is to check the converse of Pythagoras theorem. If the triangle satisfy the following formula then it will be a right angle triangle.

$\mathtt{( Side\ 3)^{2} =\ ( side\ 1)^{2} \ +\ ( side\ 2)^{2} \ }\\\ \\ \mathtt{5^{2} =\ 4^{2} \ +\ 3^{2}}\\\ \\ \mathtt{25\ =\ 16\ +\ 9}\\\ \\ \mathtt{25\ =\ 25}$

Since the triangle satisfies the above expression, we can conclude that it is a right triangle.

(02) Check if the below triangle is right angled triangle.

Solution
Let’s check if the triangle satisfy Pythagoras expression.

Here, longest side = 7 cm
Other sides are 6 cm and 5 cm long.

$\mathtt{( Side\ 3)^{2} =\ ( side\ 1)^{2} \ +\ ( side\ 2)^{2} \ }\\\ \\ \mathtt{7^{2} =\ 6^{2} \ +\ 5^{2}}\\\ \\ \mathtt{49\ =\ 36\ +\ 25}\\\ \\ \mathtt{49\ \not{=} \ 61}$

Since the Pythagoras theorem is not satisfied, the given triangle is not a right triangle.

(03) The sides of a triangle are 5 cm,12 cm and 13 cm. Check if the triangle is right angled or not.

Solution
You don’t need any diagram to check if the triangle is right angled or not.

Just check if the sides satisfy the Pythagoras expression.

$\mathtt{( Longest\ side)^{2} =\ ( side\ 1)^{2} \ +\ ( side\ 2)^{2} \ }\\\ \\ \mathtt{13^{2} =\ 12^{2} \ +\ 5^{2}}\\\ \\ \mathtt{169\ =\ 144\ +\ 25}\\\ \\ \mathtt{169\ =\ 169}$

The above expression is satisfied.
Hence, the given triangle is a right angled triangle.

Pythagoras theorem review