NCERT Solutions Class 9 Maths Chapter 12 — Heron's Formula

Solution:

Step 1: The standard formula for the area of a triangle is 1/2 × base × height. This requires knowing the height corresponding to a specific base.

Step 2: Heron's Formula, Area = √[s(s-a)(s-b)(s-c)], allows calculating the area solely from the lengths of the three sides (a, b, c) and the semi-perimeter (s).

Step 3: Therefore, it is most useful when the height is not readily available or difficult to calculate, but all three side lengths are known.

Solution:

Step 1: The perimeter of a triangle is the sum of its three sides: P = a + b + c.

Step 2: Given side lengths a = 7 cm, b = 8 cm, c = 9 cm, the perimeter P = 7 + 8 + 9 = 24 cm.

Step 3: The semi-perimeter (s) is half of the perimeter: s = P / 2. [s = (7 + 8 + 9) / 2 = 24 / 2]

Step 4: Therefore, s = 12 cm.

Solution:

Step 1: According to the triangle inequality theorem, the sum of any two sides of a triangle must be greater than the third side (e.g., a + b > c).

Step 2: If a + b > c, then a + b + c > 2c, which means 2s > 2c, or s > c. Similarly, s > a and s > b.

Step 3: This implies that (s-a), (s-b), and (s-c) must all be positive values. If any of these terms were zero or negative, it would mean the sides cannot form a valid triangle.

Step 4: The area calculated using Heron's formula, being a physical quantity, must always be positive.

Solution:

Step 1: Given sides a=5 cm, b=12 cm, c=13 cm.

Step 2: Step 1: Calculate semi-perimeter (s). s = (a + b + c) / 2 = (5 + 12 + 13) / 2 = 30 / 2 = 15 cm. Ravi's s = 15 cm is correct.

Step 3: Step 2: Calculate (s-a), (s-b), (s-c). [s-a = 15-5 = 10 cm s-b = 15-12 = 3 cm s-c = 15-13 = 2 cm]

Step 4: Ravi's calculations for (s-a), (s-b), (s-c) are correct.

Step 5: Step 3: Apply Heron's Formula: Area = √[s(s-a)(s-b)(s-c)]. [Area = √(15 × 10 × 3 × 2) = √(30 × 30) = √(900) = 30 cm².]

Step 6: Ravi's multiplication and square root calculation are also correct. Therefore, he made no mistake.

Solution:

Step 1: Let the sides be a = 10 cm, b = 17 cm, c = 21 cm.

Step 2: Step 1: Calculate the semi-perimeter (s). [s = (a + b + c) / 2 = (10 + 17 + 21) / 2 = 48 / 2 = 24 cm.]

Step 3: Step 2: Calculate (s-a), (s-b), (s-c). [s-a = 24 - 10 = 14 cm s-b = 24 - 17 = 7 cm s-c = 24 - 21 = 3 cm]

Step 4: Step 3: Apply Heron's Formula: Area = √[s(s-a)(s-b)(s-c)]. [Area = √(24 × 14 × 7 × 3) = √(2 × 12 × 2 × 7 × 7 × 3) = √(2 × 2 × 6 × 2 × 7 × 7 × 3) = √(2^4 × 3^2 × 7^2) = 2^2 × 3 × 7 = 4 × 21 = 84 cm².]