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 ↓


Similarity of Triangles


Geometry is a fascinating world of shape, measurement, and structure. One of the fundamental concepts in geometry is similarity, specifically the similarity of triangles. Understanding how triangles can be similar involves recognizing patterns and relationships between their angles and sides. In this exposition, we will learn about the criteria and underlying principles that determine when triangles are similar, supported with both drawn and textual examples.

What is triangle similarity?

When we say that two triangles are similar, we mean that they have the same shape, but not necessarily the same size. Formally, two triangles are similar if their corresponding angles are equal and their corresponding sides are in proportion. The concept of similarity is written using the symbol "~", so if triangle ABC is similar to triangle DEF, we write it as ΔABC ~ ΔDEF.

Conditions for triangle similarity

There are three main conditions or criteria that help us determine if two triangles are similar. These conditions are angle-angle (AA), side-side-side (SSS), and side-angle-side (SAS).

Angle-Angle (AA) Criterion

The angle-angle (AA) criterion for similarity asserts that if two angles of one triangle are similar to two angles of another triangle, then the triangles are similar.

ABCDIF

Consider triangles ΔABC and ΔDEF. If we know that ∠A = ∠D and ∠B = ∠E, then by the AA criterion, ΔABC ~ ΔDEF.

Side-Side-Side (SSS) Criterion

The Side-Side-Side (SSS) Criterion states that if the corresponding sides of two triangles are in proportion, then the triangles are similar.

If AB/DE = BC/EF = AC/DF,
So ΔABC ~ ΔDEF.

Example: Let AB = 4, BC = 6, AC = 8; and DE = 2, EF = 3, DF = 4 We can check:

AB/DE = 4/2 = 2,
BC/EF = 6/3 = 2,
AC/DF = 8/4 = 2.

Since all ratios are equal, by SSS criterion, ΔABC ~ ΔDEF.

Side-Angle-Side (SAS) Criterion

The Side-Angle-Side (SAS) Criterion for similarity tells us that if two sides of a triangle are in the same proportion as two sides of another triangle, and the included angles are equal, then the triangles are similar.

If AB/DE = AC/DF and ∠A = ∠D,
So ΔABC ~ ΔDEF.

Example: Let AB = 6, AC = 9, and DE = 3, DF = 4.5, also ∠A = ∠D = 45°.

AB/DE = 6/3 = 2,
AC/DF = 9/4.5 = 2.

With ∠A = ∠D, the SAS criterion confirms that ΔABC ~ ΔDEF.

Applications of triangle similarity

Understanding triangle similarity is not just an academic exercise; it has practical applications in many fields. These applications include solving problems in architecture, engineering, astronomy, and even art.

Height measurement

Equality can be used to measure the height of objects that are difficult to measure directly. For example, to determine the height of a tree, you can measure the length of its shadow and compare it to the length of the shadow cast by a smaller known height.

Suppose that a stick 1.5 meters high casts a shadow 2 meters long, while at the same time the shadow of a tree is 10 meters long. From the triangle similarity:

Height of tree / Shadow of tree = Height of stick / Shadow of stick
Height of tree / 10 = 1.5 / 2

Solving this gives the height of the tree = (1.5/2) * 10 = 7.5 meters.

Scale models of the real world

Scale models are miniature representations of larger structures, such as buildings or ships. By ensuring that all measurements are kept proportionate, the model remains a smaller version of the larger object.

Properties of similar triangles

There are many remarkable properties of similar triangles that can be used to solve a variety of geometric problems.

Compatible height

The corresponding altitudes of similar triangles are proportional to the corresponding sides.

Corresponding medians

The corresponding medians of similar triangles are proportional to the corresponding sides.

Perimeter and area

If triangles are similar, then their perimeters are proportional to the sides. The ratio of the areas of two similar triangles is equal to the square of the ratio of their corresponding sides.

If ΔABC ~ ΔDEF and AB/DE = k,
So (Area of ΔABC) / (Area of ΔDEF) = k².

Pythagorean theorem in similar triangles

In the context of similarity, the Pythagorean Theorem provides a useful tool for understanding triangular relationships. If a series of smaller triangles within a larger right triangle are similar to each other and to the larger triangle, then the Pythagorean Theorem is true on multiple levels in these triangles.

Consider a right triangle ΔABC with ∠C = 90°. A perpendicular is drawn from the vertex C to the hypotenuse at point D, forming three right triangles - ΔACD, ΔBCD, and the larger ΔABC. The three triangles are similar.

Proofs involving equality

Throughout geometry, proofs involving similar triangles can provide meaningful insights and beautiful solutions to complex problems.

Finding parallel lines with triangle similarity

If a transversal intersects two lines and makes corresponding angles that are equal, then by the converse of the corresponding angles postulate, the lines are parallel.

Consider triangles ΔABC and ΔABD, where ∠BAC = ∠BAD and AB is the side of each other. If AC is parallel to BD, then by the Angle-Angle-Side (AAS) condition any transverse line will form similar triangles in which corresponding equal angles prove the lines to be parallel.

Conclusion

Similarity of triangles is a cornerstone topic in geometry that underlies a range of practical applications and theoretical problems. By understanding these principles, students gain powerful tools for analyzing geometric shapes and concepts foundational to advanced mathematics and related subjects.


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Criteria for Similarity
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Similarity of Triangles

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