Quick Revision Notes

Class 9 - Mathematics

Chapter: Heron's Formula (हीरोन का सूत्र)

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### 1. Introduction to Triangles (त्रिभुज का परिचय)

A triangle is a closed figure bounded by three line segments. It has three sides, three angles, and three vertices.

• Basic Area Formula: $$\text{Area of a Triangle} = \frac{1}{2} \times \text{Base} \times \text{Height}$$ $$\text{(त्रिभुज का क्षेत्रफल} = \frac{1}{2} \times \text{आधार} \times \text{ऊँचाई)}$$ Note: This formula is mainly used when the height (altitude) of the triangle is known (e.g., Right-angled triangles, Equilateral triangles).

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### 2. Types of Triangles Based on Sides (भुजाओं के आधार पर त्रिभुज के प्रकार)

1. Scalene Triangle (विषमबाहु त्रिभुज): All three sides are of different lengths ($a \neq b \neq c$).
2. Isosceles Triangle (समद्विबाहु त्रिभुज): Any two sides are equal ($a = b$).
3. Equilateral Triangle (समबाहु त्रिभुज): All three sides are equal ($a = b = c$).

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### 3. Heron's Formula (हीरोन का सूत्र)

When the lengths of all three sides of a triangle are given, and it is difficult or impossible to find the height easily, Heron's Formula is used to calculate the area.

Step 1: Find the Semi-Perimeter ($s$) (अर्ध-परिमाप ज्ञात करना)

If $a$, $b$, and $c$ are the lengths of the sides of the triangle, then the semi-perimeter $s$ is given by: $$s = \frac{a + b + c}{2}$$

Step 2: Apply Heron's Formula (हीरोन का सूत्र लागू करना)

$$\text{Area of Triangle (क्षेत्रफल)} = \sqrt{s(s - a)(s - b)(s - c)}$$

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### 4. Special Cases (विशेष स्थितियाँ)

A. Area of an Equilateral Triangle (समबाहु त्रिभुज का क्षेत्रफल)

If each side of an equilateral triangle is $a$, then using Heron's formula:

• Semi-perimeter ($s$) = $\frac{a + a + a}{2} = \frac{3a}{2}$
• $\text{Area} = \frac{\sqrt{3}}{4} \times (\text{Side})^2 = \frac{\sqrt{3}}{4}a^2$

B. Area of an Isosceles Triangle (समद्विबाहु त्रिभुज का क्षेत्रफल)

If the equal sides are $a$ and the base is $b$:

• $\text{Area} = \frac{b}{4}\sqrt{4a^2 - b^2}$

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### 5. Application to Quadrilaterals (चतुर्भुजों के क्षेत्रफल में अनुप्रयोग)

To find the area of a quadrilateral (like a field or park) when all four sides and one diagonal are given:

1. Divide the quadrilateral into two triangles by drawing a diagonal.
2. Apply Heron's Formula separately to both triangles.
3. Add the areas of the two triangles to get the total area of the quadrilateral.

$$\text{Total Area} = \text{Area of Triangle 1} + \text{Area of Triangle 2}$$

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### 6. Important Tips for MP Board Exam

• Units: Always write the final answer in square units (e.g., $\text{cm}^2$, $\text{m}^2$) because it represents an area.
• Calculation: First find $s$, then find the values of $(s-a)$, $(s-b)$, and $(s-c)$ before putting them inside the square root. Look for pairs of prime factors inside the square root to simplify easily without multiplying large numbers.