Grade 9
  1. Number Systems  1.1. Real Numbers  1.2. Rational Numbers  1.3. Irrational Numbers  1.4. Properties of Real Numbers  1.5. Laws of Exponents and Surds  1.6. Representation on the Number Line  1.7. Decimal Representation of Real Numbers
  2. Polynomials  2.1. Definition and Types of Polynomials  2.2. Degree of a Polynomial  2.3. Zeroes of a Polynomial  2.4. Remainder Theorem  2.5. Factorization of Polynomials  2.6. Algebraic Identities  2.7. Polynomial Equations and Roots
  3. Coordinate Geometry  3.1. Cartesian System  3.2. Plotting Points on the Plane  3.3. Distance Formula  3.4. Section Formula  3.5. Area of a Triangle  3.6. Coordinates of Midpoint and Centroid
  4. Linear Equations in Two Variables  4.1. Solutions of a Linear Equation  4.2. Graphical Method  4.3. Algebraic Method  4.4. Applications of Linear Equations  4.5. Linear Inequalities
  5. Introduction to Euclidean Geometry  5.1. Basic Definitions and Terms  5.2. Axioms and Postulates  5.3. Theorems on Angles and Lines  5.4. Properties of Parallel Lines  5.5. Construction of Triangles  5.6. Congruence Axioms
  6. Lines and Angles  6.1. Angle Pair Relationships  6.2. Parallel Lines and Transversal  6.3. Properties of Angles Formed by Parallel Lines  6.4. Angle Sum Property of a Triangle  6.5. Types of Angles
  7. Triangles  7.1. Congruence of Triangles  7.2. Criteria for Congruence  7.3. Properties of Triangles  7.4. Inequalities in a Triangle  7.5. Similarity of Triangles  7.6. Pythagorean Theorem
  8. Quadrilaterals  8.1. Properties of Quadrilaterals  8.2. Types of Quadrilaterals  8.3. Midpoint Theorem  8.4. Properties of Parallelograms  8.5. Trapezium and Kite Properties  8.6. Diagonals and their Properties
  9. Areas of Parallelograms and Triangles  9.1. Area of a Parallelogram  9.2. Area of a Triangle  9.3. Proofs of Area Theorems  9.4. Applications of Area Theorems
  10. Circles  10.1. Definitions and Terms  10.2. Properties of a Circle  10.3. Chord Properties  10.4. Tangents to a Circle  10.5. Cyclic Quadrilaterals  10.6. Arcs and Angles Subtended
  11. Constructions  11.1. Basic Constructions  11.2. Constructing Bisectors  11.3. Constructing Triangles  11.4. Constructing Tangents to a Circle  11.5. Constructing Angles of Given Measure
  12. Heron's Formula  12.1. Area of a Triangle Using Heron’s Formula  12.2. Application to Different Triangles and Quadrilaterals  12.3. Proof of Heron’s Formula
  13. Surface Areas and Volumes  13.1. Surface Area of a Cube Cuboid and Cylinder  13.2. Volume of a Cube Cuboid and Cylinder  13.3. Surface Area and Volume of a Cone  13.4. Surface Area and Volume of a Sphere and Hemisphere  13.5. Conversion of Units  13.6. Volume of a Prism
  14. Statistics  14.1. Collection of Data  14.2. Presentation of Data  14.3. Measures of Central Tendency  14.4. Mean Median and Mode  14.5. Graphical Representation of Data  14.6. Range and Measures of Dispersion
  15. Probability  15.1. Introduction to Probability  15.2. Experimental Probability  15.3. Theoretical Probability  15.4. Simple Problems on Probability

Grade 9 → Coordinate Geometry ↓


Area of a Triangle


Introduction

Understanding the area of a triangle is very important in geometry. In coordinate geometry, finding the area of a triangle requires knowing the coordinates of its vertices. This method is useful when the vertices are placed on a coordinate plane. We use algebraic expressions involving the coordinates of the vertices to find the area without needing to know the angles or side lengths.

Formula

If you know the coordinates of the vertices of a triangle, you can find its area using the following formula:

Area = 0.5 * |x1(y2 - y3) + x2(y3 - y1) + x3(y1 - y2)|

In this formula, (x1, y1), (x2, y2) and (x3, y3) are the coordinates of the three vertices of the triangle. The vertical bars represent the absolute value, which ensures that the area is always positive.

Understanding the formula

Let's dig deeper into this formula. The idea comes from the determinant of the matrix formed by the coordinates. This formula essentially calculates twice the signed area of a triangle by considering the inclination of the lines connecting the points.

Visual example 1

Consider a triangle with vertices at (1, 2), (4, 5) and (6, 2). Let's calculate the area using the formula:

Area = 0.5 * |1(5 - 2) + 4(2 - 2) + 6(2 - 5)| = 0.5 * |1(3) + 4(0) + 6(-3)| = 0.5 * |3 + 0 - 18| = 0.5 * |-15| = 0.5 * 15 = 7.5

Therefore, the area of the triangle is 7.5 square units.

(1, 2)(4, 5)(6, 2)

Advantages of coordinate method

Using coordinates offers a lot of versatility, especially in fields such as computer graphics, physics, and engineering. It allows you to find an area without needing perpendicular heights or dividing the triangle into smaller segments. This method is rooted in algebra, making it a favorite calculation in many applications.

More examples

Example 2

Suppose a triangle has vertices (2, 3), (5, 11) and (12, 8). Calculate the area using the formula.

Area = 0.5 * |2(11 - 8) + 5(8 - 3) + 12(3 - 11)| = 0.5 * |2(3) + 5(5) + 12(-8)| = 0.5 * |6 + 25 - 96| = 0.5 * |-65| = 0.5 * 65 = 32.5

The area of this triangle is 32.5 square units.

Example 3

For a triangle with vertices (-1, 0), (3, 4) and (5, -2) the area is calculated as follows:

Area = 0.5 * |-1(4 - (-2)) + 3(-2 - 0) + 5(0 - 4)| = 0.5 * |-1(6) + 3(-2) + 5(-4)| = 0.5 * |-6 - 6 - 20| = 0.5 * |-32| = 0.5 * 32 = 16

Therefore, the area of the triangle is 16 square units.

(-1, 0)(3, 4)(5, -2)

Practice problems

Try to calculate the areas of the following triangles using the coordinates of their vertices:

  1. The vertices are at (0, 0), (4, 0) and (0, 3).
  2. The vertices are at (-3, -4), (0, 0) and (3, -4).
  3. The vertices are at (-2, 2), (3, -1) and (5, 5).

Use the formula given earlier to find the area and verify your answers.

Conclusion

In short, calculating the area of a triangle in coordinate geometry is efficient and simple when you have the coordinates of the vertices. By applying the formula, you can determine areas in a variety of applications, benefiting from this geometric insight deeply rooted in algebra. This aspect of coordinate geometry serves many purposes in both theoretical and practical contexts.


Grade 9 → 3.5


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Area of a Triangle

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