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 ↓


Similar Figures


Similar figures are fundamental concepts in geometry. In simple terms, similar figures have the same shape but not necessarily the same size. When working with these figures, we often compare their lengths and angles to understand the idea of similarity. In this comprehensive guide, we will explore what similar figures are, how to identify them, and various key concepts related to similarity.

Definition of similar shapes

Similar figures in geometry are figures that have the same shape but may vary in size. Two figures are considered similar if:

  • All corresponding angles are equal.
  • The lengths of their corresponding sides are in proportion.

Understanding these two key features will help you determine if two shapes are similar.

Proportionality of corresponding sides

An essential property of similar figures is that the ratio of the lengths of their corresponding sides is the same. To illustrate this point, let us consider two rectangles, rectangle A and rectangle B.

A B

In the example above, if the side lengths of rectangle A are in a 1:2 ratio compared to the lengths of rectangle B, then these rectangles are similar. For example, if the side lengths of rectangle A are 4 and 8, then the corresponding side lengths of rectangle B must be 8 and 16.

    Ratio of Corresponding Side Lengths: (Side 1 of A) / (Side 1 of B) = (Side 2 of A) / (Side 2 of B) 4 / 8 = 8 / 16 = 1/2
    Ratio of Corresponding Side Lengths: (Side 1 of A) / (Side 1 of B) = (Side 2 of A) / (Side 2 of B) 4 / 8 = 8 / 16 = 1/2

Equality of corresponding angles

Apart from proportional sides, another important aspect is that corresponding angles of similar figures are always equal. This is easy to see when we take two triangles, triangle X and triangle Y.

X Y

Suppose triangle X has angles of 30°, 60°, and 90°, and triangle Y also has angles of 30°, 60°, and 90°. Based on the angle criterion, these two triangles are similar.

Working with similar shapes

When working with similar figures, understanding the proportionality of sides and equality of angles helps solve a variety of geometric problems, such as determining missing side lengths or angles.

Example 1: Finding the missing side

Consider two similar triangles, triangle P and triangle Q. Triangle P has sides of length 3, 4, and 5. Triangle Q has two sides of length 6 and 8, and the third side is unknown.

    Triangle P: 3, 4, 5 Triangle Q: 6, 8, x To find x (missing side length of Triangle Q): Ratio of known sides: 3/6 = 4/8 = 5/x 1/2 = 1/2 = 5/x Cross-multiply to find x: 5 * 2 = xx = 10
    Triangle P: 3, 4, 5 Triangle Q: 6, 8, x To find x (missing side length of Triangle Q): Ratio of known sides: 3/6 = 4/8 = 5/x 1/2 = 1/2 = 5/x Cross-multiply to find x: 5 * 2 = xx = 10

Example 2: Finding the missing angle

Let us take two similar quadrilaterals, quadrilateral R and quadrilateral S. The angles of quadrilateral R are 50°, 60°, 80°, and 170°, while the angles of quadrilateral S are 50°, 60°, and 80°, with one angle missing.

    Quadrilateral R: 50°, 60°, 80°, 170° Quadrilateral S: 50°, 60°, 80°, y Since these quadrilaterals are similar, their corresponding angles are equal: Therefore, y = 170°
    Quadrilateral R: 50°, 60°, 80°, 170° Quadrilateral S: 50°, 60°, 80°, y Since these quadrilaterals are similar, their corresponding angles are equal: Therefore, y = 170°

Applications of similar shapes

Understanding analogous shapes is not just important theoretically; it also has practical applications in everyday life, such as in art, architecture, and engineering. For example, architects use analogous shapes to create scale models of buildings, helping them visualize designs before construction.

Examples in architecture

Suppose an architect designs a house and builds a model with the dimensions reduced by a factor of 1:100. The front of the real house is 200 feet wide, while the model is only 2 feet wide. These two figures are identical, obeying the same principles discussed above.

Examples in art

Artists often use the same shapes to create paintings in different sizes. Many artworks maintain proportionality of size, even when enlarged or reduced.

Properties of similar shapes

In addition to proportionality and congruence of angles, other properties come into play when dealing with similar figures:

  • Perimeter: The ratio of the perimeters of two similar figures is equal to the ratio of the lengths of their corresponding sides.
  • Area: The ratio of the areas of similar figures is the square of the ratio of the lengths of their corresponding sides. If the ratio of the lengths of the sides is r, then the ratio of the areas is r2.

Example: Calculating perimeter and area

Consider two identical squares, square F with a side of 5 units, and square G with a side of 10 units:

  • Perimeter of square F = 4 * 5 = 20
  • Perimeter of square G = 4 * 10 = 40
  • Ratio of perimeter: 20/40 = 1/2
  • Area of square F = 5 * 5 = 25
  • Area of square G = 10 * 10 = 100
  • Area ratio: 25/100 = 1/4

Testing for equality

When given two geometric shapes, several methods can be used to test their similarity:

  • AA (Angle-Angle) Criterion: If two angles of one figure are equal to two angles of another figure, then the figures are similar.
  • SSS (Side-Side-Side) Criterion: If the lengths of all corresponding sides of two figures are in proportion, then the figures are similar.
  • SAS (Side-Angle-Side) Criterion: If two sides of one figure are in the same proportion as two sides of another figure, and the angles between them are equal, then the figures are similar.

Example of equivalence testing with AA criterion

Suppose we have two triangles, where triangle A has angles of 45° and 90°, and triangle B has angles of 45° and 90°. According to the AA criterion, these triangles are similar.

Example of equivalence testing with SSS criterion

Consider two triangles, triangle C with sides 3, 4, and 5, and triangle D with sides 6, 8, and 10. The sides are in proportion:

    Ratio of corresponding sides: 3/6 = 4/8 = 5/10 = 1/2
    Ratio of corresponding sides: 3/6 = 4/8 = 5/10 = 1/2

Since the ratios are equal, the triangles are similar according to the SSS criterion.

Conclusion

Similar shapes are the cornerstone of understanding geometry. Identifying similar patterns allows mathematicians and various professionals to solve geometric problems and apply these concepts to real-world situations. By understanding the properties and criteria of similarity, you can recognize these shapes and work with them effectively.


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