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 ↓


Coordinates of Midpoint and Centroid


Coordinate geometry, also called analytical geometry, is where geometry meets algebra. Using graphs and coordinates, we can provide an extra dimension to geometry problems and find things like distances, midpoints, and centroids.

Midpoint of a line segment

The midpoint is the point that is exactly in the middle of the endpoints of the line segment. If you think about it, when you have a line segment connecting two points, there is always a point that is equidistant from both of these points. That is the midpoint.

Midpoint formula

Suppose you have a line segment with endpoints (x1, y1) and (x2, y2). The formula for the midpoint M(x, y) of a line segment is given by the average of the x- coordinate and y- coordinate of the endpoints. The formula is as follows:

    M(x, y) = ( (x1 + x2) / 2, (y1 + y2) / 2)

Example 1: Finding the midpoint

Let us calculate the midpoint of the line segment with endpoints (2, 3) and (4, 7).

    x = (2 + 4) / 2 = 3 y = (3 + 7) / 2 = 5

Thus, the midpoint is M (3, 5).

Example 2: Visual representation

Consider two points A(1, 2) and B(7, 8). Let's find the midpoint and visualize it as well:

<svg width="300" height="300" xmlns="http://www.w3.org/2000/svg"> <line x1="50" y1="250" x2="250" y2="150" stroke="black" stroke-width="2"/> <circle cx="50" cy="250" r="5" fill="red"/> <text x="35" y="265" font-family="sans-serif" font-size="16px">A(1,2)</text> <circle cx="250" cy="150" r="5" fill="blue"/> <text x="255" y="150" font-family="sans-serif" font-size="16px">B(7,8)</text> <circle cx="150" cy="200" r="5" fill="green"/> <text x="155" y="215" font-family="sans-serif" font-size="16px">M(4,5)</text> </svg>

In the above example, the line segment AB is shown with points A and B marked in red and blue, respectively. The green point is the midpoint M(4, 5) calculated using the midpoint formula.

Centroid of a triangle

The centroid is a concurrency point of the medians of a triangle. In other words, it is the place where the three medians of a triangle intersect. The median is a line segment that joins the vertex to the midpoint of the opposite side.

Formula of centroid

The centroid G(x, y) for a triangle with vertices (x1, y1), (x2, y2), and (x3, y3) is given by:

    G(x, y) = ( (x1 + x2 + x3) / 3, (y1 + y2 + y3) / 3)

Example 3: Finding the centroid

Consider a triangle with vertices (1, 2), (3, 4) and (5, 6). Let us find its centroid:

    x = (1 + 3 + 5) / 3 = 3 y = (2 + 4 + 6) / 3 = 4

Hence the centroid is G (3, 4).

Example 4: Visual example

Imagine a triangle containing the points A(2, 2), B(4, 6) and C(6, 2). We want to find and show the centroid:

<svg width="300" height="300" xmlns="http://www.w3.org/2000/svg"> <polygon points="100,200 150,100 200,200" fill="none" stroke="black" stroke-width="2"/> <circle cx="100" cy="200" r="5" fill="red"/> <text x="85" y="215" font-family="sans-serif" font-size="16px">A</text> <circle cx="150" cy="100" r="5" fill="blue"/> <text x="155" y="95" font-family="sans-serif" font-size="16px">B</text> <circle cx="200" cy="200" r="5" fill="green"/> <text x="205" y="215" font-family="sans-serif" font-size="16px">C</text> <circle cx="150" cy="167" r="5" fill="purple"/> <text x="155" y="172" font-family="sans-serif" font-size="16px">G</text> </svg>

The red point is A, the blue point is B, the green point is C, and the purple point is the centroid G.

Properties and applications

Midpoint properties

Midpoint is very useful for finding the center point between any two given points. It is widely used in the following scenarios:

  • Bisection of line segments: The midpoint divides a line segment into two equal parts.
  • Symmetry: In geometric shapes like rectangles and rhombuses the diagonals intersect at the midpoints.
  • Construction: In construction and design, midpoints can help ensure balance and symmetry.

Centroid properties

The nucleus is important in a variety of fields, especially in physics and engineering:

  • Center of mass: The centroid serves as the balance point or center of mass for a triangle of uniform density.
  • Intersection of medians: It gives a point where three medians of a triangle meet, and divides each median into segments in the ratio of 2:1.
  • Applications in the real world: It is used in finding the optimal point in networks and strategic planning.

More examples

Example 5: Midpoint in the coordinate plane

You have two points on the coordinate plane, (-3, -4) and (1, 2). What is the midpoint?

    x = (-3 + 1) / 2 = -1 y = (-4 + 2) / 2 = -1

Therefore, the midpoint is (-1, -1).

Example 6: Centroid with negative coordinate

Let us find the centroid of the triangle with vertices (-2, -3), (1, 5) and (4, -1).

    x = (-2 + 1 + 4) / 3 = 1 y = (-3 + 5 - 1) / 3 = (1) / 3 ≈ 0.33

Hence the centroid G is approximately (1, 0.33).

Conclusion

The concepts of midpoint and centroid are fundamental in coordinate geometry. They serve as stepping stones to more advanced topics and are applied in a range of practical and theoretical problems.

Understanding how to find these points through their formulas prepares you to handle and reason about more complex geometric and algebraic problems. The midpoint adds the idea of balancing a line, while the centroid extends these ideas to the world of triangles.

With these concepts in mind, one can explore complex geometric structures and applications, such as those found in trigonometry, calculus, and real-life modeling scenarios. From constructing buildings to designing animations, midpoints and centroids provide a stable framework and strategy!


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Coordinates of Midpoint and Centroid

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