NCERT Solutions Class 8 Maths Chapter 9 — Pythagorean Theorem

Solution:

Step 1: The Pythagorean theorem establishes a fundamental relationship between the three sides of a right-angled triangle.

Step 2: It states that the square of the hypotenuse (the side opposite the right angle) is equal to the sum of the squares of the other two sides (legs). [a² + b² = c²]

Step 3: Therefore, its application is strictly limited to right-angled triangles.

Solution:

Step 1: The Pythagorean theorem states a² + b² = c², where 'c' is the hypotenuse (the longest side).

Step 2: Given two sides of a right-angled triangle, the unknown side could be either the hypotenuse or one of the legs.

Step 3: If 5 cm and 12 cm are the legs, then the equation should be 5² + 12² = x² (where x is the hypotenuse). In this case, x would be √(25+144) = √169 = 13 cm.

Step 4: If 12 cm is the hypotenuse and 5 cm is one leg, then the equation 5² + x² = 12² is correct for finding the other leg. Ravi's mistake is in not considering the possibility that 12 cm could be the hypotenuse if x is the other leg, implying x is smaller than 12.

Solution:

Step 1: In a right-angled triangle, the angle measuring 90° is called the right angle.

Step 2: The side directly opposite to the right angle is defined as the hypotenuse.

Step 3: Since the right angle is at Q, the side opposite to angle Q is PR. Therefore, PR is the hypotenuse.

Solution:

Step 1: A Pythagorean triplet consists of three positive integers that satisfy the Pythagorean theorem.

Step 2: If 'a', 'b', and 'c' are natural numbers, they form a Pythagorean triplet if a² + b² = c².

Step 3: This means they can be the sides of a right-angled triangle, where 'c' is the hypotenuse.

Solution:

Step 1: When a diagonal is drawn in a rectangle, it forms two right-angled triangles.

Step 2: The length and width of the rectangle act as the two legs (perpendicular and base) of the right-angled triangle, and the diagonal is the hypotenuse.

Step 3: Using the Pythagorean theorem: Diagonal² = Length² + Width².

Step 4: Diagonal² = 80² + 60² = 6400 + 3600 = 10000. So, Diagonal = √10000 = 100 meters.