This MCQ module is based on: 6.1 Introduction
TOPIC 14 OF 35
6.1 Introduction
🎓 Class 10
Mathematics
CBSE
Theory
Ch 6 — Triangles
⏱ ~15 min
🌐 Language: [gtranslate]
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This mathematics assessment will be based on: 6.1 Introduction
Targeting Class 10 level in Geometry, with Intermediate difficulty.
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6.1 Introduction
In earlier classes, you studied congruence of figures — when two figures have exactly the same shape and the same size. In this chapter, we relax the "same size" requirement and study similarity?: figures having the same shape but not necessarily the same size. A photograph and its enlargement, two maps of the same region at different scales, and the Sun and a coin during a solar eclipse — all illustrate similarity.
Fig 6.1: Triangles of the same shape but different sizes — intuitively "similar"
6.2 Similar Figures
Observe that all circles with the same radius are congruent, and all circles (irrespective of radius) have the same shape — they are similar. The same is true of all squares, and all equilateral triangles.
Definition
Two polygons of the same number of sides are similar if(i) their corresponding angles are equal, and
(ii) their corresponding sides are in the same ratio (i.e., proportional).
Both conditions are necessary for polygons of four or more sides. A rectangle and a square have equal corresponding angles but their sides are not proportional — they are not similar. A rhombus and a square have proportional sides but unequal angles — again not similar.
Symbol: "Triangle ABC is similar to triangle DEF" is written as \(\triangle ABC \sim \triangle DEF\). The order of vertices is critical: A ↔ D, B ↔ E, C ↔ F.
Similarity Ratio
If \(\triangle ABC \sim \triangle DEF\), then \(\angle A = \angle D\), \(\angle B = \angle E\), \(\angle C = \angle F\), and \(\tfrac{AB}{DE} = \tfrac{BC}{EF} = \tfrac{CA}{FD} = k\), the scale factor.
Fig 6.2: Quadrilateral ABCD ~ quadrilateral PQRS with equal corresponding angles and proportional sides
Historical Note — Thales of Miletus
Thales (640–546 BCE), one of the seven sages of ancient Greece, reportedly measured the height of the Great Pyramid of Giza by comparing its shadow to the shadow of his staff. Because the Sun's rays are parallel, the triangles formed by the pyramid and its shadow, and by the staff and its shadow, are similar. This was one of the earliest recorded uses of similarity.
6.3 Similarity of Triangles
Triangles are a special case: for triangles, proportionality of sides alone implies equality of angles (and vice versa). But the full definition remains both conditions simultaneously.
The cornerstone result for triangle similarity is the Basic Proportionality Theorem (BPT), also called Thales' Theorem, first proved rigorously by Thales.
Theorem 6.1 — Basic Proportionality Theorem (BPT)
If a line is drawn parallel to one side of a triangle, intersecting the other two sides in distinct points, then it divides those two sides in the same ratio.That is: in \(\triangle ABC\), if DE ∥ BC with D on AB and E on AC, then \(\dfrac{AD}{DB} = \dfrac{AE}{EC}\).
Fig 6.3: DE ∥ BC ⇒ AD/DB = AE/EC (BPT)
Proof of BPT
Proof
Construction: Join BE and CD, and draw DM ⊥ AC and EN ⊥ AB.Step 1: \(\text{ar}(\triangle ADE) = \tfrac12 \cdot AD \cdot EN\); \(\text{ar}(\triangle BDE) = \tfrac12 \cdot DB \cdot EN\).
So \(\dfrac{\text{ar}(\triangle ADE)}{\text{ar}(\triangle BDE)} = \dfrac{AD}{DB}\).
Step 2: Similarly, \(\dfrac{\text{ar}(\triangle ADE)}{\text{ar}(\triangle DEC)} = \dfrac{AE}{EC}\).
Step 3: \(\triangle BDE\) and \(\triangle DEC\) stand on the same base DE and lie between the same parallels DE ∥ BC. Hence \(\text{ar}(\triangle BDE) = \text{ar}(\triangle DEC)\).
Conclusion: From Steps 1, 2, 3: \(\dfrac{AD}{DB} = \dfrac{AE}{EC}\). ∎
Theorem 6.2 — Converse of BPT
If a line divides any two sides of a triangle in the same ratio, then the line is parallel to the third side.In \(\triangle ABC\), if D lies on AB and E on AC with \(\tfrac{AD}{DB}=\tfrac{AE}{EC}\), then DE ∥ BC.
Example 1 — Applying BPT
In \(\triangle ABC\), DE ∥ BC. If AD = 1.5 cm, DB = 3 cm, and AE = 1 cm, find EC.
By BPT, \(\tfrac{AD}{DB}=\tfrac{AE}{EC}\Rightarrow \tfrac{1.5}{3}=\tfrac{1}{EC}\Rightarrow EC = 2\) cm.
Example 2 — Using Converse BPT
In \(\triangle PQR\), S and T are points on PQ and PR respectively with PS = 4 cm, SQ = 4.5 cm, PT = 8 cm, TR = 9 cm. Is ST ∥ QR?
\(\tfrac{PS}{SQ} = \tfrac{4}{4.5} = \tfrac{8}{9}\); \(\tfrac{PT}{TR} = \tfrac{8}{9}\). Ratios equal ⇒ by Converse BPT, ST ∥ QR.
Example 3 — BPT with algebra
In \(\triangle ABC\), DE ∥ BC, AD = \(x\), DB = \(x-2\), AE = \(x+2\), EC = \(x-1\). Find \(x\).
\(\tfrac{x}{x-2} = \tfrac{x+2}{x-1}\Rightarrow x(x-1) = (x+2)(x-2)\Rightarrow x^2 - x = x^2 - 4\Rightarrow x = 4\).
Example 4 — Midpoint Theorem as a corollary
If D and E are midpoints of AB and AC respectively, then \(\tfrac{AD}{DB} = 1 = \tfrac{AE}{EC}\), so by Converse BPT, DE ∥ BC. Further, DE is half of BC. (This is the familiar Midpoint Theorem.)
Activity: Verify BPT by Construction
L3 ApplyMaterials: Paper, ruler, pencil, a set square or protractor.
Predict: If you draw any line parallel to BC inside a triangle ABC, will it cut AB and AC in equal ratios?
- Draw any triangle ABC with base BC at the bottom.
- Using a set square, draw a line DE parallel to BC, with D on AB and E on AC.
- Measure AD, DB, AE, EC carefully in cm.
- Compute \(\tfrac{AD}{DB}\) and \(\tfrac{AE}{EC}\). Compare.
- Repeat with a different parallel line and a different triangle.
Within measurement error, the two ratios are always equal. This confirms the Basic Proportionality Theorem experimentally.
Exercise 6.2 (Selected)
Q1. In \(\triangle ABC\), DE ∥ BC. (i) AD = 1.5, DB = 3, AE = 1, find EC. (ii) AD = 4, AE = 8, DB = \(x-4\), EC = \(3x-19\). Find \(x\).
(i) EC = 2 (see Example 1).
(ii) \(\tfrac{4}{x-4} = \tfrac{8}{3x-19}\Rightarrow 4(3x-19)=8(x-4)\Rightarrow 12x-76=8x-32\Rightarrow 4x=44\Rightarrow x=11\).
(ii) \(\tfrac{4}{x-4} = \tfrac{8}{3x-19}\Rightarrow 4(3x-19)=8(x-4)\Rightarrow 12x-76=8x-32\Rightarrow 4x=44\Rightarrow x=11\).
Q2. E and F are points on sides PQ and PR of \(\triangle PQR\). Determine if EF ∥ QR: (i) PE = 3.9, EQ = 3, PF = 3.6, FR = 2.4. (ii) PE = 4, QE = 4.5, PF = 8, RF = 9.
(i) \(\tfrac{PE}{EQ}=\tfrac{3.9}{3}=1.3\); \(\tfrac{PF}{FR}=\tfrac{3.6}{2.4}=1.5\). Unequal, so EF ∦ QR.
(ii) \(\tfrac{4}{4.5}=\tfrac{8}{9}\). Equal, so EF ∥ QR.
(ii) \(\tfrac{4}{4.5}=\tfrac{8}{9}\). Equal, so EF ∥ QR.
Q3. In \(\triangle ABC\), if LM ∥ CB and LN ∥ CD, prove that \(\tfrac{AM}{AB}=\tfrac{AN}{AD}\).
In \(\triangle ACB\), LM ∥ CB ⇒ \(\tfrac{AM}{MB}=\tfrac{AL}{LC}\). In \(\triangle ACD\), LN ∥ CD ⇒ \(\tfrac{AN}{ND}=\tfrac{AL}{LC}\). So \(\tfrac{AM}{MB}=\tfrac{AN}{ND}\). Adding 1 to both sides: \(\tfrac{AB}{MB}=\tfrac{AD}{ND}\), and inverting: \(\tfrac{AM}{AB}=\tfrac{AN}{AD}\). ∎
Q4. Using BPT, prove that a line drawn through the midpoint of one side of a triangle parallel to another side bisects the third side (Midpoint Theorem converse).
Let D be the midpoint of AB, and let DE ∥ BC with E on AC. By BPT, \(\tfrac{AD}{DB}=\tfrac{AE}{EC}\). Since AD=DB, we get AE=EC, so E bisects AC. ∎
Competency-Based Questions
Scenario: A photographer places a vertical 1 m ruler and a tall tree side by side on level ground. At the same time, the shadow of the ruler is 1.5 m and the shadow of the tree is 12 m.
Q1. Using similarity, find the height of the tree.
L3 Apply(b) 8 m. Ratio: \(\tfrac{h}{1}=\tfrac{12}{1.5}\Rightarrow h=8\) m.
Q2. Analyse why the two right-triangles (ruler-shadow and tree-shadow) must be similar.
L4 AnalyseBoth triangles are right-angled at the base, and the Sun's rays make the same angle of elevation with the ground at that instant (because the Sun is effectively at infinity). Two angles equal ⇒ by AA similarity, the triangles are similar.
Q3. Evaluate: if a student measured the tree's shadow 5 minutes later as 13.5 m with ruler shadow still 1.5 m, which reading would give the correct height?
L5 EvaluateShadows of the ruler and tree must be taken at the same instant (same Sun angle). The ruler shadow of 1.5 m was measured first; if 5 min later the Sun has moved, both shadows must be re-measured together. Mixing measurements from different times gives a wrong answer. The method is valid only with simultaneous shadows.
Q4. Design a method to measure the height of a school flagpole using only a metre ruler and a clear day. State the assumptions and calculation.
L6 CreateStand the 1 m ruler vertically next to the flagpole at a time when both cast clear shadows on level ground. At the same instant, measure the ruler's shadow length \(s\) and the flagpole's shadow length \(S\). By similar triangles, height \(H = \tfrac{S}{s}\times 1\) m. Assumptions: ground level, pole vertical, same-instant measurements.
Assertion–Reason Questions
Assertion (A): All squares are similar.
Reason (R): All squares have corresponding angles equal (90°) and sides in ratio 1:1.
Reason (R): All squares have corresponding angles equal (90°) and sides in ratio 1:1.
Answer: (a).
Assertion (A): A rectangle and a square are always similar.
Reason (R): Both have all angles equal to 90°.
Reason (R): Both have all angles equal to 90°.
Answer: (d). A is false because rectangle sides needn't be in 1:1 ratio. R is true.
Assertion (A): In \(\triangle ABC\), if DE ∥ BC with D on AB, E on AC, and AD = 2, DB = 3, AE = 4, then EC = 6.
Reason (R): BPT.
Reason (R): BPT.
Answer: (a). \(\tfrac{2}{3}=\tfrac{4}{EC}\Rightarrow EC=6\).
Frequently Asked Questions — Triangles
What is Part 1 — Similar Figures & BPT (Thales' Theorem) | Class 10 Maths Ch 6 | MyAiSchool in NCERT Class 10 Mathematics?
Part 1 — Similar Figures & BPT (Thales' Theorem) | Class 10 Maths Ch 6 | MyAiSchool is a key concept covered in NCERT Class 10 Mathematics, Chapter 6: Triangles. This lesson builds the student's foundation in the chapter by explaining the core ideas with worked examples, definitions, and step-by-step methods aligned to the CBSE curriculum.
How do I solve problems on Part 1 — Similar Figures & BPT (Thales' Theorem) | Class 10 Maths Ch 6 | MyAiSchool step by step?
To solve problems on Part 1 — Similar Figures & BPT (Thales' Theorem) | Class 10 Maths Ch 6 | MyAiSchool, follow the NCERT method: identify the given quantities, choose the relevant formula or theorem, substitute values carefully, and simplify. Class 10 exercises gradually increase in difficulty — start with solved NCERT examples before attempting exercise questions, and always verify your answer by substitution or diagram.
What are the most important formulas for Chapter 6: Triangles?
The essential formulas of Chapter 6 (Triangles) are listed in the chapter summary and highlighted throughout the lesson in formula boxes. Memorise them and practise at least 2–3 problems per formula. CBSE board exams frequently test direct application as well as combined use of multiple formulas from this chapter.
Is Part 1 — Similar Figures & BPT (Thales' Theorem) | Class 10 Maths Ch 6 | MyAiSchool important for the Class 10 board exam?
Part 1 — Similar Figures & BPT (Thales' Theorem) | Class 10 Maths Ch 6 | MyAiSchool is part of the NCERT Class 10 Mathematics syllabus and appears in CBSE board exams. Questions typically include short-answer, long-answer, and competency-based items. Review the NCERT examples, exercise questions, and previous-year board problems on this topic to prepare confidently.
What mistakes should students avoid in Part 1 — Similar Figures & BPT (Thales' Theorem) | Class 10 Maths Ch 6 | MyAiSchool?
Common mistakes in Part 1 — Similar Figures & BPT (Thales' Theorem) | Class 10 Maths Ch 6 | MyAiSchool include skipping steps, misapplying formulas, sign errors, and losing track of units. Write each step clearly, double-check algebraic manipulations, and re-read the question after solving to verify that your answer matches what was asked.
Where can I find more NCERT practice questions on Part 1 — Similar Figures & BPT (Thales' Theorem) | Class 10 Maths Ch 6 | MyAiSchool?
End-of-chapter NCERT exercises for Part 1 — Similar Figures & BPT (Thales' Theorem) | Class 10 Maths Ch 6 | MyAiSchool cover all difficulty levels tested in CBSE exams. After completing them, try the examples again without looking at the solutions, attempt the NCERT Exemplar questions for Chapter 6, and solve at least one previous-year board paper to consolidate your understanding.
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