How to Identify Right Triangles Using the Converse of Pythagoras
What is the Converse of Pythagoras Theorem?
If the square of a side is equal to the sum of the squares of the other two sides then the triangle must be a right angle triangle, this is known to be the Converse of Pythagoras Theorem. This is the Converse of Pythagoras statement. The Pythagorean theorem in Mathematics states that the sum of the square of two sides (legs) is equal to the square of the hypotenuse of a right-angle triangle. But, in the reverse of the Pythagorean theorem, it is known that if this relation satisfies, then the triangle must be a right angle triangle. So, if the sides of a triangle have length, a, b and c and satisfy the given condition c²= a² + b², then the triangle is known to be a right-angle triangle.
What is the Pythagoras Theorem?
Consider a right-angle triangle ABC, with its three sides namely the opposite, adjacent and the hypotenuse. In a right-angled triangle we generally refer to the three sides in order to their relation with the angle θ. The little box in the right corner of the triangle given below denotes the right angle which is equal to 90°.
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The side opposite to the right angle is the longest side of the triangle which is known as the hypotenuse(H). The side that is opposite to the angle θ is known as the opposite(O). And the side which lies next to the angle θ is known as the Adjacent(A)
The Pythagoras theorem states that,
Converse of Pythagorean Theorem Proof:
The converse of the Pythagorean Theorem proof is:
Converse of Pythagoras theorem statement: The Converse of Pythagoras theorem statement says 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 of a triangle, then the triangle is known to be a right triangle.
That is, in ΔABC if c²= a² + b² then
∠C is a right triangle, the ΔPQR being the right angle.
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We can prove this by contradiction.
Let us assume that ,
c²= a² + b² in ΔABC and the triangle is not a right triangle.
Now consider another triangle ΔPQR. We construct ΔPQR so that
PR=a, QR=b and ∠R is a right angle.
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By the Pythagorean Theorem,
(PQ)² = a² + b²
But we know that , a² + b² = c² and c=AB.
So, (PQ)² = a² + b² = (AB)²
That is, (PQ)² = (AB)²
Since the lengths of the sides are PQ and AB, we can take positive square roots.
PQ=AB
That is, all the three sides of the triangle PQR are congruent to the three sides of the triangle ABC. So, the two triangles are congruent by the Side-Side-Side Congruence Property.
Since ΔABC is congruent to ΔPQR and ΔPQR is a right triangle, ΔABC must also be a right triangle.
This is a contradiction. Therefore, our assumption must be wrong.This is the converse of Pythagoras theorem proof.
Formula of Converse of the Pythagorean Theorem:
As per the converse of the Pythagorean theorem, the formula for a right-angled triangle is given by:
Where the variables a, b and c are the sides of a triangle.
Applications of the converse of Pythagoras theorem:
Basically, the converse of the Pythagoras theorem is used to find whether the measurements of a given triangle belong to the right triangle or not. If we come to know that the given sides belong to a right-angled triangle, it helps in the construction of such a triangle. Using the concept of the converse of Pythagoras theorem, one can determine if the given three sides form a Pythagorean triplet.
Questions to be Solved:
Question 1) Check whether the triangle with the side lengths, 5, 7 and 9 units is an acute ,right or an obtuse triangle.
Answer) The longest side of the triangle has a length equal to 9 units. Now, compare the square of the length of the longest side with the sum of squares of the other two sides that have been given.
Square of the length of the longest side is 9² = 81 units.
Sum of the squares of the other two sides is equal to,
5² + 7² is equal to 25+ 49 which is equal to 74 square units.
This concludes that, 9² > 5² + 7²
Therefore, by the corollary to the converse of the Pythagorean Theorem, the triangle is an obtuse triangle.
Question 2) Check whether a triangle with side lengths 6 cm, 10 cm, and 8 cm is a right triangle. Check whether the square of the length of the longest side is the sum of the squares of the other two sides.
Answer) Apply the converse of Pythagorean Theorem.
(10)² = (6)² + (8)²
100 = 36 + 64
Since the square of the length of the longest side is the sum of the squares of the other two sides, by the converse of the Pythagorean Theorem, the triangle is referred to as a right triangle.
A corollary to the theorem categorizes the triangles into acute triangle, right triangle, or obtuse triangle.
In a triangle with side lengths a, b, and c where c is the length of the longest side,
If (c)² < (a)² + (b)² then we can say that the triangle is acute, and
If (c)² < (a)² + (b)² then we can say that the triangle is obtuse.
FAQs on Converse of Pythagoras Theorem Explained
1. What is the official statement of the Converse of the Pythagoras Theorem?
The Converse of the Pythagoras Theorem states that if in a triangle, the square of the length of the longest side is equal to the sum of the squares of the other two sides, then the angle opposite the longest side is a right angle. Essentially, if a triangle with sides a, b, and c (where c is the longest side) satisfies the condition a² + b² = c², then it must be a right-angled triangle.
2. What is the main difference between the Pythagoras Theorem and its converse?
The main difference lies in the hypothesis and conclusion. The Pythagoras Theorem starts with a known right-angled triangle and concludes a relationship between its sides (a² + b² = c²). In contrast, the Converse of the Pythagoras Theorem starts with a known relationship between the sides (a² + b² = c²) and concludes that the triangle must be right-angled. One proves a property of sides from a known angle, while the other proves the nature of an angle from a known property of sides.
3. How do you apply the Converse of the Pythagoras Theorem to check if a triangle is right-angled?
To apply the theorem, follow these steps:
Step 1: Identify the lengths of the three sides of the triangle.
Step 2: Find the longest side. Let its length be 'c'. The other two sides will be 'a' and 'b'.
Step 3: Calculate the square of the longest side (c²).
Step 4: Calculate the sum of the squares of the other two sides (a² + b²).
Step 5: Compare the results. If c² = a² + b², the triangle is a right-angled triangle.
4. Why is the Converse of the Pythagoras Theorem important in real-world applications?
Its importance lies in its ability to verify right angles without needing an angle-measuring tool. In fields like construction, carpentry, and architecture, workers can ensure that corners are perfectly square (90 degrees) by simply measuring the lengths of the sides of a triangle. For example, to check if a wall corner is a right angle, they can measure 3 units along one wall, 4 units along the other, and check if the diagonal distance between those two points is exactly 5 units (since 3² + 4² = 5²).
5. What if the square of the longest side is NOT equal to the sum of the squares of the other two sides?
This is a key extension of the theorem used to classify triangles. Let 'c' be the longest side:
If a² + b² > c², the triangle is an acute-angled triangle.
If a² + b² < c², the triangle is an obtuse-angled triangle.
This shows that the relationship between the side squares can determine not only if a triangle is right-angled, but also whether it is acute or obtuse.
6. Can you show a numerical example of applying the Converse of the Pythagoras Theorem?
Certainly. Consider a triangle with side lengths of 8 cm, 15 cm, and 17 cm. To check if it is a right-angled triangle:
The longest side (c) is 17 cm.
Square of the longest side: c² = 17² = 289.
Sum of the squares of the other two sides: a² + b² = 8² + 15² = 64 + 225 = 289.
Since 289 = 289, the condition is met. Therefore, the triangle with sides 8 cm, 15 cm, and 17 cm is a right-angled triangle.
7. What is a common mistake students make when using the Converse of the Pythagoras Theorem?
A very common mistake is not identifying the longest side correctly before applying the formula. The theorem's condition a² + b² = c² is only valid if 'c' represents the longest side of the triangle. If a student randomly assigns the side lengths to a, b, and c without ensuring 'c' is the largest value, their conclusion about the triangle being right-angled will be incorrect.
8. What are the key steps to prove the Converse of the Pythagoras Theorem as per the Class 10 CBSE syllabus for 2025-26?
The proof involves construction and congruence. The main steps are:
Given: A triangle ΔABC where AC² = AB² + BC².
Construction: Construct another triangle ΔPQR, which is right-angled at Q, such that its sides PQ = AB and QR = BC.
Proof using Pythagoras Theorem: In ΔPQR, since ∠Q = 90°, we have PR² = PQ² + QR². By construction, this becomes PR² = AB² + BC².
Comparison: We are given AC² = AB² + BC². From our construction, we found PR² = AB² + BC². This implies AC² = PR², so AC = PR.
Congruence: Now compare ΔABC and ΔPQR. We have AB = PQ, BC = QR (by construction), and AC = PR (as proved). Thus, ΔABC ≅ ΔPQR by the SSS (Side-Side-Side) congruence rule.
Conclusion: Since the triangles are congruent, their corresponding parts are equal. Therefore, ∠B = ∠Q. As ∠Q = 90°, we conclude that ∠B = 90°, proving the theorem.
