Mastering Triangles Ex 6.5 Class 10 NCERT Maths: Pythagoras Theorem and its Converse

Revisiting Pythagoras Theorem and its Converse

The world of geometry is built upon fundamental principles, and among them, the Pythagoras Theorem stands as a monumental pillar, especially when dealing with right-angled triangles. Discovered by the ancient Greek mathematician Pythagoras, this theorem provides a direct relationship between the lengths of the sides of a right triangle. It 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." If we denote the two shorter sides (legs) as 'a' and 'b', and the longest side (hypotenuse) as 'c' (always opposite the 90-degree angle), the theorem can be expressed as a² + b² = c². This formula is indispensable for finding the length of an unknown side when the other two sides are known, making it a cornerstone for various calculations in engineering, architecture, and even everyday life, from designing ramps to calculating distances, proving its immense practical relevance.

Equally important, and often overlooked, is the Converse of Pythagoras Theorem. While the original theorem tells us a property of right triangles, its converse allows us to identify whether a triangle is indeed right-angled. The converse 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 the first side is a right angle." This means if you have a triangle with sides x, y, and z, and you find that x² + y² = z², then the angle opposite side z must be 90 degrees. This powerful tool is used to prove the existence of right angles within more complex geometric figures, which can then unlock further properties and solutions. For instance, if you're given three side lengths of a triangle, you can use the converse to determine its type—acute, obtuse, or right-angled. Furthermore, understanding the geometric proof of Pythagoras Theorem, often involving similar triangles formed by an altitude to the hypotenuse, deepens your conceptual grasp of how these relationships are derived. Mastering both the theorem and its converse comprehensively is key to excelling in Exercise 6.5 and building a strong foundation for future mathematical concepts.

Key Definitions

Right-angled Triangle

A triangle in which one of the interior angles measures exactly 90 degrees.

Hypotenuse

The side opposite the right angle in a right-angled triangle. It is always the longest side of the triangle.

Legs (of a right triangle)

The two sides that form the right angle in a right-angled triangle. These are also often referred to as the perpendicular and the base.

Pythagorean Triplet

A set of three positive integers (a, b, c) such that a² + b² = c². Common examples include (3, 4, 5) and (5, 12, 13).

Step-by-Step: Applying Pythagoras Theorem and its Converse

1. For Finding a Missing Side (using Pythagoras Theorem) — If you are given a right-angled triangle and the lengths of two sides, follow these steps to find the third: 1. Identify the Right Angle and Hypotenuse: First, locate the 90-degree angle. The side directly opposite this angle is the hypotenuse (c), which is always the longest side. The other two sides are the legs (a and b). 2. State the Formula: Write down the Pythagoras Theorem: a² + b² = c². 3. Substitute Known Values: Carefully plug in the lengths of the two known sides into the formula. Remember to square each value. 4. Solve for the Unknown Side: Perform the necessary algebraic operations (addition, subtraction, and taking the square root) to determine the length of the missing side. Ensure your final answer is simplified and includes appropriate units.
2. For Verifying a Right Triangle (using Converse of Pythagoras Theorem) — If you are given the lengths of all three sides of a triangle and need to determine if it is a right-angled triangle, follow these steps: 1. Identify the Longest Side: Among the three given side lengths, identify the longest one. This side will be your potential hypotenuse (let's call its length 'c'). Let the other two shorter sides be 'a' and 'b'. 2. Calculate the Square of the Longest Side: Compute the square of the length of the longest side (c²). 3. Calculate the Sum of Squares of the Other Two Sides: Square the lengths of the two shorter sides (a² and b²) and then add these squared values together (a² + b²). 4. Compare the Results: Now, compare the result from Step 2 (c²) with the result from Step 3 (a² + b²). If c² = a² + b², then the triangle is a right-angled triangle. If c² ≠ a² + b², then the triangle is not a right-angled triangle. 5. Conclude: State your conclusion clearly. If it is a right-angled triangle, also mention that the right angle is opposite the side you identified as 'c' (the longest side).

Worked Examples

• Example 1: Finding the Hypotenuse Q: In a right-angled triangle ABC, right-angled at B, if AB = 6 cm and BC = 8 cm, find the length of AC. A: Step 1: Identify the knowns and unknown. Given: Triangle ABC is right-angled at B. AB = 6 cm (leg), BC = 8 cm (leg). To find: AC (hypotenuse). Step 2: Apply the Pythagoras Theorem. According to Pythagoras Theorem, in ΔABC, AC² = AB² + BC². Step 3: Substitute the values. AC² = (6)² + (8)² = 36 + 64 = 100. Step 4: Solve for AC. AC = √100 = 10 cm. Final Answer: The length of AC is 10 cm.
• Example 2: Verifying a Right Triangle Q: The sides of a triangle are 7 cm, 24 cm, and 25 cm. Determine if it is a right-angled triangle. A: Step 1: Identify the longest side. The given sides are 7 cm, 24 cm, and 25 cm. The longest side is 25 cm. Let c = 25 cm, a = 7 cm, and b = 24 cm. Step 2: Calculate the square of the longest side (c²). c² = (25)² = 625. Step 3: Calculate the sum of the squares of the other two sides (a² + b²). a² + b² = (7)² + (24)² = 49 + 576 = 625. Step 4: Compare the results. Since c² = 625 and a² + b² = 625, we have c² = a² + b². Step 5: Conclude using the Converse of Pythagoras Theorem. By the Converse of Pythagoras Theorem, the triangle is a right-angled triangle, and the angle opposite the side of length 25 cm is the right angle. Final Answer: Yes, the triangle with sides 7 cm, 24 cm, and 25 cm is a right-angled triangle.
• Example 3: Real-World Application Q: A ladder 15 m long leans against a vertical wall. If the foot of the ladder is 9 m away from the base of the wall, how high up the wall does the ladder reach? A: Step 1: Visualize the problem. The wall, the ground, and the ladder form a right-angled triangle, where the wall is perpendicular to the ground. The ladder is the hypotenuse, the distance from the wall to the foot of the ladder is one leg, and the height the ladder reaches on the wall is the other leg. Step 2: Identify knowns and unknown. Length of ladder (hypotenuse) = 15 m. Distance from wall (base) = 9 m. Let the height up the wall be 'h' (perpendicular). Step 3: Apply the Pythagoras Theorem. h² + 9² = 15². Step 4: Substitute and solve for 'h'. h² + 81 = 225. Subtract 81 from both sides: h² = 225 - 81 = 144. Step 5: Calculate 'h'. h = √144 = 12 m. Final Answer: The ladder reaches 12 meters high up the wall.

Exam Tips & Common Mistakes

To score well in questions related to Triangles Exercise 6.5, keep these points in mind and avoid common pitfalls:

• Confusing Hypotenuse: Always correctly identify the hypotenuse. It is always the side opposite the 90-degree angle and is the longest side of a right-angled triangle. Mistakenly using a leg as the hypotenuse is a frequent error.
• Applicability: Remember that the Pythagoras Theorem and its Converse are exclusively for right-angled triangles. Do not attempt to apply them to acute or obtuse triangles, as the results will be incorrect.
• Algebraic Accuracy: Pay close attention to your calculations. Errors in squaring numbers, adding/subtracting, or taking square roots can lead to incorrect final answers. Double-check your arithmetic, especially with multi-step problems.
• Clear Conclusion for Converse: When using the Converse of Pythagoras Theorem to prove a triangle is right-angled, explicitly state which angle is 90 degrees (it's the angle opposite the side whose square equals the sum of the squares of the other two sides).
• Units and Simplification: Always include appropriate units (cm, m, etc.) in your final answer. If the answer involves a square root that isn't a perfect square, simplify it to its simplest radical form unless otherwise specified (e.g., √50 = 5√2).

Practice Questions with Solutions

• Q: In a right-angled triangle ABC, right-angled at B, AB = 7 cm and BC = 24 cm. Find the length of AC. A: Step 1: Identify the knowns and unknown. Given a right-angled triangle ABC at B, with legs AB = 7 cm and BC = 24 cm. We need to find the hypotenuse AC. Step 2: Apply the Pythagoras Theorem. According to Pythagoras Theorem, AC² = AB² + BC². Step 3: Substitute the given values. AC² = (7)² + (24)² = 49 + 576 = 625. Step 4: Solve for AC. AC = √625 = 25 cm. Final answer: The length of AC is 25 cm.
• Q: Determine if a triangle with sides 9 cm, 40 cm, and 41 cm is a right-angled triangle. A: Step 1: Identify the longest side. The longest side is 41 cm. Let a = 9 cm, b = 40 cm, c = 41 cm. Step 2: Calculate the square of the longest side (c²). c² = (41)² = 1681. Step 3: Calculate the sum of the squares of the other two sides (a² + b²). a² + b² = (9)² + (40)² = 81 + 1600 = 1681. Step 4: Compare the results. Since c² = 1681 and a² + b² = 1681, we have c² = a² + b². Step 5: Conclude. By the Converse of Pythagoras Theorem, the triangle is a right-angled triangle. The right angle is opposite the side of length 41 cm. Final answer: Yes, the triangle is a right-angled triangle.
• Q: A 13 m long wire has been attached to a vertical pole of height 5 m. The other end of the wire is attached to the ground. How far from the base of the pole should the point on the ground be, so that the wire is taut? A: Step 1: Visualize the setup as a right-angled triangle. The pole forms the perpendicular, the ground distance forms the base, and the wire forms the hypotenuse. Step 2: Identify knowns and unknown. Height of pole (perpendicular) = 5 m. Length of wire (hypotenuse) = 13 m. Let the distance from the base of the pole to the point on the ground be 'x' (base). Step 3: Apply the Pythagoras Theorem. (Hypotenuse)² = (Perpendicular)² + (Base)². So, 13² = 5² + x². Step 4: Substitute and solve for x. 169 = 25 + x². x² = 169 - 25 = 144. Step 5: Calculate x. x = √144 = 12 m. Final answer: The point on the ground should be 12 m away from the base of the pole.
• Q: In an isosceles triangle ABC, AC = BC. If AB² = 2AC², prove that ΔABC is a right-angled triangle. A: Step 1: Analyze the given information. We are given an isosceles triangle ABC where AC = BC. We are also given the relation AB² = 2AC². Step 2: Substitute the isosceles property into the given relation. Since AC = BC, we can rewrite 2AC² as AC² + AC². So, AB² = AC² + AC². Step 3: Further substitute using the isosceles property. As AC = BC, we can replace one AC² with BC². This gives AB² = AC² + BC². Step 4: Apply the Converse of Pythagoras Theorem. We have now shown that the square of one side (AB²) is equal to the sum of the squares of the other two sides (AC² + BC²). Step 5: Conclude. By the Converse of Pythagoras Theorem, ΔABC is a right-angled triangle, and the angle opposite the side AB (which is angle C) must be 90 degrees. Final answer: Since AB² = AC² + BC², by the Converse of Pythagoras Theorem, ΔABC is a right-angled triangle with ∠C = 90°.

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