Grade 10
  1. Number Systems  1.1. Real Numbers  1.1.1. Properties of Real Numbers  1.1.2. Rational and Irrational Numbers  1.1.3. Decimal Representation of Real Numbers  1.1.4. Operations on Real Numbers  1.1.5. Real Numbers on the Number Line  1.2. Euclid's Division Lemma  1.3. Prime Factorization  1.4. LCM and HCF  1.5. Exponents and Radicals  1.5.1. Laws of Exponents  1.5.2. Simplifying Radical Expressions  1.5.3. Scientific Notation
  2. Algebra  2.1. Polynomials  2.1.1. Definition and Types of Polynomials  2.1.2. Degree of a Polynomial  2.1.3. Zeros of a Polynomial  2.1.4. Factorization of Polynomials  2.1.5. Algebraic Identities  2.2. Linear Equations in Two Variables  2.2.1. Graph of Linear Equations  2.2.2. Solutions of Linear Equations  2.2.3. Application of Linear Equations in Word Problems  2.3. Quadratic Equations  2.3.1. Standard Form of a Quadratic Equation  2.3.2. Methods of Solving Quadratic Equations  2.3.2.1. Factorization Method  2.3.2.2. Completing the Square Method  2.3.2.3. Quadratic Formula  2.3.3. Nature of Roots  2.4. Arithmetic Progressions  2.4.1. Definition of Arithmetic Progression  2.4.2. General Term of an AP  2.4.3. Sum of First n Terms of an AP  2.5. Introduction to Functions  2.5.1. Definition and Types of Functions  2.5.2. Domain and Range  2.5.3. Composition of Functions  2.5.4. Inverse of a Function
  3. Coordinate Geometry  3.1. Cartesian System  3.2. Distance Formula  3.3. Section Formula  3.4. Area of a Triangle  3.5. Slope of a Line  3.6. Equation of a Line  3.6.1. Slope-Intercept Form  3.6.2. Point-Slope Form  3.6.3. Two-Point Form  3.6.4. Intercept Form  3.6.5. General Form of a Line
  4. Trigonometry  4.1. Trigonometric Ratios  4.1.1. Sine Cosine Tangent and their Reciprocals  4.1.2. Values of Trigonometric Ratios at Specific Angles  4.2. Trigonometric Identities  4.3. Trigonometric Ratios of Complementary Angles  4.4. Heights and Distances  4.4.1. Angle of Elevation and Depression  4.4.2. Applications in Real-Life Problems
  5. Geometry  5.1. Triangles  5.1.1. Congruence of Triangles  5.1.2. Criteria for Congruence  5.1.3. Properties of Triangles  5.2. Similarity  5.2.1. Similar Figures  5.2.2. Criteria for Similarity  5.2.3. Similarity of Triangles  5.2.4. Applications of Similarity in Real-Life  5.3. Circles  5.3.1. Properties of a Circle  5.3.2. Tangent to a Circle  5.3.3. Number of Tangents from a Point  5.3.4. Angle Subtended by an Arc  5.4. Constructions  5.4.1. Construction of Similar Triangles  5.4.2. Tangent to a Circle  5.4.3. Construction of Angle Bisectors
  6. Mensuration  6.1. Areas of Plane Figures  6.1.1. Area of a Triangle  6.1.2. Area of a Parallelogram  6.1.3. Area of a Trapezium  6.1.4. Area of a Rhombus  6.2. Surface Areas and Volumes  6.2.1. Surface Area of a Cube Cuboid and Cylinder  6.2.2. Surface Area of a Cone and Sphere  6.2.3. Volume of a Cube Cuboid and Cylinder  6.2.4. Volume of a Cone and Sphere  6.3. Conversion of Units  6.4. Frustum of a Cone
  7. Statistics  7.1. Introduction to Statistics  7.2. Collection of Data  7.3. Presentation of Data  7.3.1. Tabular Form  7.3.2. Graphical Form  7.3.2.1. Bar Graph  7.3.2.2. Histogram  7.3.2.3. Frequency Polygon  7.3.2.4. Ogive  7.4. Measures of Central Tendency  7.4.1. Mean Median and Mode  7.4.2. Quartiles and Percentiles  7.5. Measures of Dispersion  7.5.1. Range and Interquartile Range  7.5.2. Standard Deviation and Variance
  8. Probability  8.1. Introduction to Probability  8.2. Experimental Probability  8.3. Theoretical Probability  8.4. Probability of Simple Events  8.5. Complementary Events  8.6. Probability of Compound Events

Grade 10 → Geometry ↓


Similarity


In geometry, "similarity" describes a special relationship between two shapes. When two shapes are similar, it means they have the same shape, but they can differ in size. In other words, one shape can be a smaller version of another. This concept is fundamental in geometry, helping to understand the properties and relationships of shapes without worrying about their actual shapes.

Why are the two shapes similar?

For two geometric shapes to be similar, they must satisfy two main conditions:

  1. Corresponding angles are equal. This means that each angle in one figure is the same as the measure of its corresponding angle in the other figure.
  2. Corresponding sides are proportional. The ratio between the lengths of corresponding sides must be constant. This means that if a side in one figure is twice as long, the corresponding side in the other figure must also be twice as long, and so on.

Understanding these situations allows us to detect similarities in various geometric shapes such as triangles, rectangles, circles, and more complex shapes.

Visual example: similar triangles

Triangles are the simplest polygons, and they often serve as the starting point for exploring geometric similarities. Let's consider two triangles:

Triangle A: Sides 3, 4, and 5
Triangle B: sides 6, 8, and 10

These triangles are similar because:

  1. Their corresponding angles are equal.
  2. The sides of triangle B are exactly twice the length of triangle A. This means that the shape of the triangles remains the same, but the size differs.
Ratio of sides:

    AB (3/6) = 1/2
    BC (4/8) = 1/2
    CA (5/10) = 1/2

Each pair of corresponding side lengths maintains the same ratio, which confirms the similarity of the triangles.

ABCA'B'C'

Exploring proportionality with similar shapes

When discussing equality, proportionality is an important element. Let's understand this concept with rectangles.

Example: Similar rectangles

Consider two rectangles, rectangle X and rectangle Y:

Rectangle X: 2 cm by 4 cm
Rectangle Y: 4 cm by 8 cm

To find out if they are the same:

  1. Corresponding angles of rectangles are always equal because each corner of a rectangle is equal to 90 degrees.
  2. Check the proportionality of the sides:
Ratio of corresponding sides:

    Length: 2/4 = 1/2
    Width: 4/8 = 1/2

The sides maintain the same proportion, which confirms the similarity of the rectangles.

Similarity in circles

Circles are unique in the discussion of similarity. Since all circles have the same size, every circle is similar to every other circle. The size of a circle is determined by its radius, diameter, or circumference, but its shape does not change.

Example: Comparing circles

Consider two circles:

Circle 1: Radius = 3 cm
Circle 2: Radius = 6 cm

These circles are similar because all circles are similar. The only change is the measurement of their radii. The similarity ratio can also be applied to circle parameters:

Ratio of radii: 3/6 = 1/2

The main properties of similar figures

Understanding the properties of similar figures helps to identify and verify similarity in geometry. These include:

  • Angle-Angle (AA) Similarity: In triangles, being similar simply requires that two angles of one triangle are equal to two angles of the other triangle.
  • Side-Side-Side (SSS) Similarity: If the sides of two triangles are in proportion, then the triangles are similar.
  • Side-Angle-Side (SAS) Similarity: If the ratio of two corresponding sides and the included angle is equal, then the triangles are similar.

Example: Triangle similarity with Angle-Angle (AA)

Consider two triangles:

Triangle G:

Angle: 45°, 55°, 80°

Triangle H:

Angle: 45°, 55°, 80°

Since two pairs of corresponding angles are equal, triangle G is similar to triangle H by the AA similarity principle.

Applications of parallelism

Similarity is more than a theoretical concept; it has real applications in architecture, engineering, art, and other fields. It helps in creating scale models, determining unknown distances or lengths through indirect measurements, and much more.

Example: Using parallelism in real life

If you know that two buildings are similar and you know the height of one of them, you can determine the height of the other building by using the ratio of their corresponding measurements.

Building A: 125 meters high
Building B (similar, but smaller): 25 m wide, the corresponding width in B is 37.5 m

Width ratio: 25:37.5 = 2:3

Therefore the actual height of Building B will also remain in the ratio of 2:3 with the height of Building A:

125 m / x = 2/3
X = 125 * 3 / 2
x = 187.5 m

Scale models

Architects and engineers often create scale models using the principle of similarity. Such models are miniature models while maintaining the proportional size of the actual structures, providing a concrete form for understanding buildings before construction.

Conclusion

Understanding similarity in geometry involves recognizing that two figures can have the same shape but differ in size. This concept relies on equal angles and proportional sides. Visualizing these relationships helps simplify complex topics, making it an essential component in a variety of practical and academic applications.

Whether you're comparing triangles, rectangles, circles, or working on a scale model, the principles of similarity provide consistent guidelines that help with both theoretical mathematical problems and real-world scenarios.


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Similar Figures

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