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 ↓


Applications of Similarity in Real-Life


In geometry, similarity means when two figures are similar in shape but possibly different in size. Two geometric figures are similar if their corresponding angles are equal and their corresponding sides are proportional. This fundamental concept can be applied to solve many problems in everyday life. Understanding how similarity works in geometry can help us create scale models, use maps, or even understand patterns in nature.

Understanding equality through examples

Consider two triangles. If all the angles of the first triangle are equal to the corresponding angles of the second triangle, then the two triangles are similar. Let's take a closer look at this with a diagram:

Here, two triangles have the same shape but different sizes. They are similar because their corresponding angles are equal. By measuring, you can see that the angles at the vertices are equal. The sides are also in proportion, which forms the basis of their similarity.

Laws of equality

Before looking at real-life examples, let's take a look again at the rules that dictate similarity in geometry. These rules can help identify similar shapes:

  • Angle-Angle (AA): If two angles of a triangle are equal to two angles of another triangle, then the triangles are similar.
  • Side-Angle-Side (SAS): If an angle of a triangle is equal to an angle of another triangle and the sides containing these angles are in proportion, then the triangles are similar.
  • Side-Side-Side (SSS): If the sides of two triangles are in proportion, then the triangles are similar.

Scale models

One of the most common applications of likeness in real life is creating scale models. These are miniature versions of larger objects like cars, airplanes or even buildings. Architects and engineers use likenesses to create models in proportion to real structures.

For example, when designing a new skyscraper, architects work on scale models that can be as small as 1/1000 of the original size but maintain the same proportions. This helps visualise the final structure and is possible due to the principles of similarity.

Suppose the actual height of the skyscraper is 200 m.
A scale model is designed with a ratio of 1:1000.

Thus, height of the model = 200 m ÷ 1000 = 0.2 m or 20 cm.

Maps and navigation

Maps are another area where proportionality is widely used. A map is a reduced representation of an area, maintained at scale. For example, on a map, a distance of 1 cm may represent 1 km in reality. This is possible because maps use a scale to ensure that all features are represented proportionately.

Actual distance between two cities = 150 km
Scale of the map = 1 cm : 50 km

Thus, map distance = 150 km ÷ 50 km/cm = 3 cm

Photography

In photography, especially portrait and architectural photography, the concept of proportionality helps photographers keep the proportions of objects correct. Using the zoom feature enlarges the image but keeps the geometry the same, preserving the correct proportions.

Art and design

Artists and designers often use likeness to create visually appealing work. When recreating artwork or designing complementary items, proportions need to be aligned correctly. For example, a logo design may appear on different media sizes, but its basic look must remain intact through likeness.

In the example above, the yellow rectangle can be seen as part of a poster design, and the orange rectangle can be the same design on a billboard. Both are similar because their width to height ratio remains constant, creating a uniform look.

Understanding the patterns of nature

In nature, we often see similarities in patterns, such as the spirals of shells, the symmetry of leaves, and even in flowers. These patterns follow geometric similarities. For example, the nautilus shell has chambers that grow in size while maintaining their same shape. This natural phenomenon can often be seen in the Fibonacci sequence and golden ratio structures.

The use of parallelism in construction

In construction, engineers and builders use similarity for a variety of applications, such as designing roofs, bridges or ramps, where maintaining angles is important. Similar triangles ensure that structures remain safe and functional, while maintaining the proportional balance needed to hold.

Example problems

Here are some typical problems that demonstrate how parallelism can be applied in practical situations:

Example 1: Shadow measurement

You can use equality to determine the height of an object using its shadow. Consider a situation where you need to find the height of a tree:

  1. Where there is sunlight, measure the shadow of the tree by standing upright with a one meter long stick.
  2. Suppose the shadow of the tree is 9 meters, and the shadow of the stick is 1.5 meters.
Height of the stick / Length of the shadow of the stick = Height of the tree / Length of the shadow of the tree
1 m / 1.5 m = tree height / 9 m

On solving this the height of the tree = 6 m

Example 2: Construction of a ramp

A ramp should be built 1.5 m above the ground level to give access to the door for a wheel-chair. As per building codes, the slope should not exceed a ratio of 1:15.

Vertical rise (door height) = 1.5 m
Maximum slope gradient = depth of base of ramp / vertical rise  1.5 m * 15 = 22.5 m

Conclusion

Understanding similarities in geometry gives us the knowledge we need to solve real-world problems. It provides clarity in creating scale, proportion, and maintaining consistent designs in a variety of fields, including art, architecture, nature, and engineering. Recognizing and applying these principles can greatly enhance our spatial understanding and open the door to many problem-solving opportunities.


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Similarity of Triangles
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Applications of Similarity in Real-Life

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