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Propositions and their Converses - Converse of the Baudhayana-Pythagoras Theorem

Grade 9CBSE

Review the key concepts, formulae, and examples before starting your quiz.

🔑Concepts

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The Baudhayana-Pythagoras Theorem states that in a right-angled triangle, the square of the hypotenuse is equal to the sum of the squares of the other two sides: a2+b2=c2a^2 + b^2 = c^2.

Right-angled triangle ABC with sides a, b, and hypotenuse c.
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The Converse of the Baudhayana-Pythagoras Theorem states: If in a triangle, the square of one side is equal to the sum of the squares of the other two sides, then the angle opposite to the first side is a right angle.

A triangle illustrating the condition for the converse theorem.
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A Pythagorean Triple is a set of three positive integers (a,b,c)(a, b, c) such that a2+b2=c2a^2 + b^2 = c^2. Common examples include (3,4,5)(3, 4, 5), (5,12,13)(5, 12, 13), and (8,15,17)(8, 15, 17).

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To verify if a triangle with side lengths xx, yy, and zz (where zz is the longest side) is right-angled, we check if the equation x2+y2=z2x^2 + y^2 = z^2 holds true.

📐Formulae

a2+b2=c2a^2 + b^2 = c^2

If AC2=AB2+BC2 in △ABC, then ∠B=90∘\text{If } AC^2 = AB^2 + BC^2 \text{ in } \triangle ABC, \text{ then } \angle B = 90^\circ

💡Examples

Problem 1:

Check whether a triangle with sides 7 cm7\text{ cm}, 24 cm24\text{ cm}, and 25 cm25\text{ cm} is a right-angled triangle.

Triangle with sides 7, 24, and 25.

Solution:

  1. Let the sides be a=7 cma = 7\text{ cm}, b=24 cmb = 24\text{ cm}, and c=25 cmc = 25\text{ cm}.
  2. Calculate the squares of the sides: a2=72=49a^2 = 7^2 = 49 b2=242=576b^2 = 24^2 = 576 c2=252=625c^2 = 25^2 = 625
  3. Sum of the squares of the smaller sides: a2+b2=49+576=625a^2 + b^2 = 49 + 576 = 625
  4. Since a2+b2=c2a^2 + b^2 = c^2 (625=625625 = 625), the condition of the Converse of the Baudhayana-Pythagoras theorem is satisfied. Therefore, the triangle is right-angled.

Explanation:

By squaring the side lengths, we find that the sum of the squares of the two shorter sides equals the square of the longest side. According to the converse theorem, this identifies the triangle as right-angled.

Problem 2:

In △PQR\triangle PQR, PQ=8 cmPQ = 8\text{ cm}, QR=6 cmQR = 6\text{ cm}, and PR=10 cmPR = 10\text{ cm}. Find the measure of ∠Q\angle Q.

Triangle PQR with sides 8, 6, 10.

Solution:

  1. Given sides: PQ=8PQ = 8, QR=6QR = 6, PR=10PR = 10.
  2. Identify the longest side: PR=10PR = 10.
  3. Calculate squares: PQ2=82=64PQ^2 = 8^2 = 64 QR2=62=36QR^2 = 6^2 = 36 PR2=102=100PR^2 = 10^2 = 100
  4. Check the relation: PQ2+QR2=64+36=100PQ^2 + QR^2 = 64 + 36 = 100 Since PQ2+QR2=PR2PQ^2 + QR^2 = PR^2, the triangle is right-angled at the vertex opposite to the longest side PRPR.
  5. The vertex opposite to PRPR is QQ. Thus, ∠Q=90∘\angle Q = 90^\circ.

Explanation:

The side lengths satisfy the Baudhayana-Pythagorean relation a2+b2=c2a^2 + b^2 = c^2. Since PRPR is the hypotenuse, the angle opposite to it, which is ∠Q\angle Q, must be 90∘90^\circ.