The Triangles and its Properties Ex 6.5: Mastering the Pythagoras Theorem

Welcome, future math wizards! In this chapter on triangles, Exercise 6.5 introduces a very special and powerful idea: the Pythagoras Theorem. This theorem is a magical key that only works for a specific type of triangle—the right-angled triangle. Have you ever wondered how builders make sure corners are perfectly square, or how we can find the distance between two points on a map? The Pythagoras theorem is the secret behind it all!

This page will guide you through everything you need to know for The Triangles and its Properties Ex 6.5 Class 7 NCERT. We will explore what a right-angled triangle is, understand the theorem's simple formula, and practice using it to find missing side lengths. By the end, you'll be able to solve problems from your textbook with confidence and see how math is used in the world around you. Let's get started!

Key Terms for Right-Angled Triangles

Right-Angled Triangle: A triangle in which one of the three angles is exactly 90 degrees (a right angle).
Hypotenuse: The longest side of a right-angled triangle. It is always the side opposite the 90-degree angle.
Legs: The two sides of a right-angled triangle that form the right angle. They are also sometimes called the base and the perpendicular.

Understanding the Pythagoras Theorem

The Pythagoras Theorem is a fundamental rule in geometry that describes the relationship between the three sides of a right-angled triangle. The theorem is named after the ancient Greek mathematician, Pythagoras.

It states that in any right-angled triangle, the square of the length of the hypotenuse (the side opposite the right angle) is equal to the sum of the squares of the lengths of the other two sides (the legs).

If we call the lengths of the two legs 'a' and 'b', and the length of the hypotenuse 'c', the formula is:

a² + b² = c²

Imagine drawing a square on each side of the triangle. The area of the square on side 'a' would be a², the area of the square on side 'b' would be b², and the area on the hypotenuse 'c' would be c². The Pythagoras theorem tells us that the areas of the two smaller squares (on the legs) add up perfectly to equal the area of the biggest square (on the hypotenuse). This simple but powerful formula allows us to find the length of any side of a right-angled triangle if we know the lengths of the other two sides.

How to Apply the Pythagoras Theorem: Step-by-Step

Step 1: Identify the Right-Angled Triangle and its Sides — First, make sure the triangle you are working with has a 90° angle. Then, identify the three sides. The side opposite the 90° angle is the hypotenuse (c). The other two sides are the legs (a and b).
Step 2: Write Down the Pythagorean Formula — Always start by writing the formula: a² + b² = c². This helps you remember it and avoid mistakes.
Step 3: Substitute the Known Values — Look at the problem and find the lengths of the sides you know. Carefully place these numbers into the formula. For example, if you know the legs are 3 cm and 4 cm, you would write: 3² + 4² = c².
Step 4: Solve for the Unknown Side — Perform the calculations. First, find the squares of the numbers (e.g., 3² = 9, 4² = 16). Then, add or subtract as needed to find the value of the unknown square (e.g., 9 + 16 = 25, so c² = 25). Finally, find the square root to get the length of the side (e.g., √25 = 5, so c = 5 cm). If you need to find a leg (say, 'a'), you will rearrange the formula to a² = c² - b².

Watch Out! Common Mistakes to Avoid

Confusing the Hypotenuse: Always identify the hypotenuse first! It is the side opposite the right angle, and it is always the longest side. Students often incorrectly place a leg's value in the 'c' position in the formula.
Incorrect Formula: A common error is mixing up the formula, like writing a² + c² = b² or a² - b² = c². Always stick to (leg)² + (leg)² = (hypotenuse)².
Forgetting the Square Root: After you calculate a² + b² to get c², you are not done! You have found the square of the hypotenuse. The final step is to take the square root to find the actual length, 'c'.
Using it for All Triangles: The Pythagoras theorem is special! It ONLY works for right-angled triangles. Do not try to apply it to acute or obtuse triangles.

Practice Questions with Solutions

Q: PQR is a triangle, right-angled at P. If PQ = 10 cm and PR = 24 cm, find QR. A: Step 1: Identify the sides. The triangle is right-angled at P. So, the side opposite P, which is QR, is the hypotenuse. PQ and PR are the legs. Step 2: Apply the Pythagoras theorem: (PQ)² + (PR)² = (QR)². Step 3: Substitute the values: 10² + 24² = (QR)². Step 4: Calculate the squares: 100 + 576 = (QR)². Step 5: Add the values: 676 = (QR)². Step 6: Find the square root: QR = √676 = 26 cm. Final answer: The length of QR is 26 cm.
Q: A 15 m long ladder reached a window 12 m high from the ground on placing it against a wall. Find the distance of the foot of the ladder from the wall. A: Step 1: Visualize the problem. The ladder forms the hypotenuse of a right-angled triangle, the wall is one leg, and the ground is the other leg. The length of the ladder (hypotenuse) is 15 m. The height of the window (one leg) is 12 m. We need to find the distance of the foot of the ladder from the wall (the other leg). Step 2: Let the distance from the wall be 'a'. Using Pythagoras theorem: a² + 12² = 15². Step 3: Calculate the squares: a² + 144 = 225. Step 4: Isolate a² by subtracting 144 from both sides: a² = 225 - 144. Step 5: Perform the subtraction: a² = 81. Step 6: Find the square root: a = √81 = 9 m. Final answer: The distance of the foot of the ladder from the wall is 9 m.
Q: Can the sides 6 cm, 8 cm, and 10 cm form a right-angled triangle? A: Step 1: To check if the sides can form a right-angled triangle, we use the converse of the Pythagoras theorem. We check if the square of the longest side equals the sum of the squares of the other two sides. Step 2: The longest side is 10 cm. This would be the hypotenuse. The other two sides (legs) are 6 cm and 8 cm. Step 3: Check the condition: (leg)² + (leg)² = (hypotenuse)² => 6² + 8² = 10²? Step 4: Calculate the squares: 36 + 64 = 100? Step 5: Perform the addition: 100 = 100. Step 6: Since the equation is true, the sides can form a right-angled triangle. Final answer: Yes, the sides 6 cm, 8 cm, and 10 cm can form a right-angled triangle.
Q: Find the perimeter of a rectangle whose length is 40 cm and a diagonal is 41 cm. A: Step 1: A rectangle's diagonal divides it into two right-angled triangles. The diagonal is the hypotenuse, and the length and breadth are the legs. Step 2: We are given the length (leg) = 40 cm and the diagonal (hypotenuse) = 41 cm. We need to find the breadth (other leg), let's call it 'b'. Step 3: Apply Pythagoras theorem: (length)² + (breadth)² = (diagonal)². So, 40² + b² = 41². Step 4: Calculate squares: 1600 + b² = 1681. Step 5: Solve for b²: b² = 1681 - 1600 = 81. So, b = √81 = 9 cm. Step 6: The question asks for the perimeter of the rectangle. The formula for the perimeter is 2 (length + breadth). Step 7: Calculate the perimeter: 2 (40 + 9) = 2 * 49 = 98 cm. Final answer: The perimeter of the rectangle is 98 cm.

Frequently Asked Questions

What is the Pythagoras Theorem in simple words?

In any triangle with a 90-degree angle, if you square the two shorter sides and add them, the total will be the same as the square of the longest side (the hypotenuse).

Can we use the Pythagoras Theorem for any kind of triangle?

No, it is a special property that only works for right-angled triangles. You cannot use it for triangles where no angle is 90 degrees.

How do I identify the hypotenuse in a triangle?

The hypotenuse is always the side directly opposite the right angle (the 90° corner). It is also always the longest side of the right-angled triangle.

How do I check if three given side lengths form a right-angled triangle?

Identify the longest side. Square its length. Then, square the other two side lengths and add them together. If this sum equals the square of the longest side, then it is a right-angled triangle.