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 → Constructions ↓


Construction of Similar Triangles


Geometry is a fascinating area of mathematics that helps us understand the shapes and spaces around us. An important concept in geometry is the idea of similar triangles. Similar triangles are triangles that have the same shape, but they can vary in size. This means that their corresponding angles are equal, and the sides are proportional.

What are similar triangles?

Similar triangles have equal corresponding angles and proportional corresponding sides. This means that if one triangle can be made larger or smaller while maintaining the same size, then these triangles are called similar triangles. For example, if triangle ABC and triangle DEF are similar, then:

    ∠A = ∠D
    ∠B = ∠E
    ∠C = ∠F

And, the corresponding sides are in proportion as follows:

    AB/DE = BC/EF = CA/FD

Basic principles of construction

The construction of similar triangles is based on certain geometric principles. These principles allow us to construct a triangle that has the same shape as a given triangle but has a different size. The main principles used in the construction of similar triangles are given below.

Angle-Angle (AA) Criterion

If two angles of one triangle are equal to two angles of another triangle, then the triangles are similar. In geometry, this is called the angle-angle (AA) similarity criterion. It works like this:

    1. Given: Two triangles with two angles equal.
    2. Construction: Construct a triangle from the given triangles.
    3. Reproduce: Draw lines parallel to the sides of the original triangle from the endpoints of a line segment.

Visual example:

A B C D I F

In the above visual example, triangle ABC is similar to triangle DEF by the AA criterion, since their corresponding angles are equal.

Side-Angle-Side (SAS) Criterion

If one angle of a triangle is equal to one angle of another triangle and the sides including these angles are proportional, then the triangles are similar according to the Side-Angle-Side (SAS) criterion. Here is the procedure:

    1. Given: An angle of a triangle is equal to an angle of another triangle whose sides around the angles are proportional.
    2. Construction: Use the angle and a side to begin constructing a triangle.
    3. Scale: Construct sides using a proportional relationship.

For example, in triangle ABC and triangle DEF:

    ∠A = ∠D
    AB/DE = AC/DF

Side-Side-Side (SSS) Criterion

The Side-Side-Side (SSS) criterion for similar triangles suggests that if the corresponding sides of two triangles are proportional, then the two triangles are similar to each other. Here's how it's used:

    1. Given: Proportional sides of two triangles.
    2. Construction: Start with one side of the triangle first.
    3. Scale: Use proportional equality to extend or shorten the other sides.

Steps to draw similar triangles

Let's look at the process of constructing a triangle similar to the given triangle using practical steps. These steps can be followed using the traditional tools used in geometry, compass, ruler and protractor.

Step-by-step construction using the AA criterion

Consider two triangles, where you have to construct a triangle similar to the given triangle:

  1. Begin by drawing a line of any length.
  2. At one endpoint of the line, reconstruct one of the given angles of the original triangle using the protractor.
  3. At the other end, reconstruct the other angle.
  4. Extend the lines until they join together to form a triangle.

Try this:

Let us consider triangle ABC where ∠ABC = 60° and ∠BCA = 50°.

  1. Draw the line BC.
  2. Construct an angle ∠ABC = 60° at point B.
  3. Construct an angle ∠BCA = 50° at point C.
  4. Extend the lines until they meet at point A, forming a triangle.

Visual example:

A B C

Adjusting the size

To make a triangle larger or smaller while maintaining similarity, measure the lengths of the sides while keeping the angles constant.

Practical uses of similar triangles

Similar triangles are fundamental in many applications, such as surveying, model building, and art. They help create optical illusions in architecture using the principles of proportional scaling, and in engineering to design identical machines or structures.

Conclusion

Constructing similar triangles plays a vital role not just in geometry but also in diverse fields such as engineering, science and arts. Understanding how to identify and construct similar triangles using methods such as AA, SAS and SSS criteria develops foundational geometry skills needed to solve real-world problems. Practicing these constructions using simple tools increases spatial awareness and accuracy, thereby enhancing one's mathematical journey.


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Construction of Similar Triangles

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