NCERT Solutions Class 10 Maths Chapter 6 Triangles

In class 10 maths triangles, you will get to know about the similarity of triangles and apply this knowledge in giving a simple proof of Pythagoras Theorem learnt earlier. Can you guess how heights of mountains (say Mount Everest) or distances of some long distant objects (say moon) have been found out? Do you think these have been measured directly with the help of a measuring tape? In fact, all these heights and distances have been found out using the idea of indirect measurements, which is based on the principle of similarity of figures.

The NCERT Solutions for Class 10 Maths - Chapter 6 provide detailed step-by-step answers to all exercises, helping students enhance their problem-solving skills and deepen their understanding of the subject. Class 10 maths chapter 6 solutions are valuable for exam preparation, ensuring that students are well-equipped to tackle questions related to triangles in their exams.

1.0NCERT Solutions for Class 10 Maths Chapter 6 Traingles : Free PDF

Students can download the free PDF of NCERT Solutions for Class 10 Maths Chapter 6 Triangles from below:

2.0NCERT Solutions Class 10 Maths Chapter 6 Triangles: All Exercises

The exercises included in Class 10 Maths Chapter 6 Triangles are tailored to develop students' understanding of different types of triangles. Students can review their understanding of the important concepts, including congruence of triangles and properties of triangles, and the application of important theorems (e.g., the Pythagorean Theorem), by responding to these questions. As students continuously practice these exercises, students can review important information that may increase clarity, thus, helping students further develop their understanding and problem-solving capabilities.  

3.0What Will Students Learn in Chapter 6: Triangles?

• Criteria for triangle congruence such as SSS (Side-Side-Side), SAS (Side-Angle-Side), and ASA (Angle-Side-Angle).
• Understanding the properties, including the Triangle Inequality Theorem.
• Applying the Pythagorean Theorem to right-angled triangles.
• Criteria for triangle similarity, including AA (Angle-Angle), SSS (Side-Side-Side), and SAS (Side-Angle-Side) similarity criteria.
• Learning how the areas of similar triangles are proportional to the squares of their corresponding sides.
• Solving practical problems related to triangles using the learned theorems and properties.

4.0Important Topics of Class 10 Maths Chapter 6 Triangles

Chapter 6, Triangles, explores the properties of triangles and criteria for similarity in triangles, including the Basic Proportionality Theorem and Pythagoras Theorem. Students learn about the different criteria for similarity, such as AA, SSS, and SAS, and how to apply these to solve problems. The chapter also covers the relationship between the areas of similar triangles and their corresponding sides. Mastering these concepts helps in solving geometric problems and understanding the properties of triangles in depth.

1. Basic Proportionality Theorem (Thales’ Theorem)
2. Criteria for Similarity of Triangles
3. Pythagoras Theorem
4. Similar Triangles

5.0General Outline for Class 10 Maths Chapter 6 Triangles

Understanding Key Concepts:

Types of Triangles:

• Based on sides: Equilateral, Isosceles, Scalene
• Based on angles: Acute-angled, Right-angled, Obtuse-angled
• Basic Terminology: Vertices, Sides, Angles, Altitude, Median, Perpendicular bisector, etc.
• Congruence of Triangles: SAS, SSS, ASA, AAS, RHS criteria.
• Similarity of Triangles: Criteria for similarity (AA, SSS, SAS).

Important Theorems and Formulas:

1. Pythagoras Theorem: $a^2+b^2=c^2$ for a right-angled triangle.
2. Basic Proportionality Theorem (Thales' Theorem): If a line is parallel to one side of a triangle and intersects the other two sides, it divides those sides proportionally.
3. Area of Triangle: The different formulas to calculate the area of a triangle are:

• Area = $\frac{1}{2}\times \text{base}\times \text{height}$
• Heron’s formula: $\text{Area}=\sqrt{s(s-a)(s-b)(s-c)}wheres=\frac{a+b+c}{2}$ is the semi-perimeter.

Steps for Solving Triangle Problems:

Step 1: Identify the type of triangle and the given information.

Step 2: Apply relevant theorems or properties (Pythagoras, similarity, proportionality, etc.).

Step 3: Use appropriate formulas for area, perimeter, or side lengths.

Step 4: Solve for unknown values by setting up equations, applying algebraic methods.

Step 5: Verify the solution, if applicable (e.g., check triangle inequality or check for congruence/similarity).

Practical Problem Types:

• Right-Angled Triangles: Use Pythagoras theorem for side calculations or trigonometric ratios (sine, cosine, tangent).
• Similar Triangles: Use properties of proportional sides and angles to solve for unknowns.
• Congruent Triangles: Use congruence criteria to prove sides or angles are equal.
• Area Problems: Apply Heron’s formula, trigonometric formulas, or basic area calculations.

Tips for Problem Solving:

• Always draw a clear diagram based on the given data.
• Label all known and unknown values clearly.
• For congruence and similarity, always check the conditions (angle equality or proportionality).
• Use algebraic equations to solve for unknown sides or angles where necessary.

This outline provides a structured approach to solving problems related to triangles in Class 10 Maths Chapter 6 Triangles.

6.0Sample NCERT Solutions for Class 10 Maths Chapter 6 Triangles

1. Fill in the blanks using the correct word given in brackets :

(i) All circles are ________ (congruent, similar)

(ii) All squares are ________ . (similar, congruent)

(iii) All ________ triangles are similar.(isosceles, equilateral)

(iv) Two polygons of the same number of sides are similar, if (a) their corresponding angles are ___ and (b) their corresponding sides are ________ . (equal, proportional)

Solution:

(i) All circles are similar.

(ii) All squares are similar.

(iii) All equilateral triangles are similar.

(iv) Two polygons of the same number of sides are similar, if (a) their corresponding angles are equal and (b) their corresponding sides are proportional.

2. Give two different examples of pair of

(i) Similar figures.

(ii) Non-similar figures.

Solution:

(i) Examples of a pair of similar figures

1. Pair of equilateral triangles are similar figures.

2. Pair of squares are similar figures.

(ii) Examples of a pair on non-similar figures

1. One equilateral triangle and one isosceles triangle are non-similar.

2. Square and rectangle are non-similar.

3. Sides of some triangles are given below. Determine which of them are right triangles. In case of a right triangle, write the length of its hypotenuse.

(i) 7 cm, 24 cm, 25 cm

(ii) 3 cm, 8 cm, 6 cm

(iii) 50 cm, 80 cm, 100 cm

(iv) 13 cm, 12 cm, 5 cm

Solution:

(i) (7)² + (24)² = 49 + 576 = 625 = (25)²

Therefore, given sides 7 cm, 24 cm, 25 cm make a right triangle.

(ii) (6)² + (3)² = 36 + 9 = 45

(8)² = 64

(6)² + (3)² ≠ (8)²

Therefore, given sides 3 cm, 8 cm, 6 cm does not make a right triangle.

(iii) (50)² + (80)² = 2500 + 6400 = 8900

(100)² = 10000

(50)² + (80)² ≠ 100²

Therefore, given sides 50 cm, 80 cm, 100 cm does not make a right triangle.

(iv) (12)² + (5)² = 144 + 25 = 169 = (13)²

Therefore, given sides 13 cm, 12 cm, 5 cm make a right triangle.

4. PQR is a triangle with ∠PQR = ∠PRQ, and M is a point on QR such that PM ⊥ QR.

Show that: PM² = QM × MR.

Solution:

∠1 + ∠2 = ∠2 + ∠4  (Each is 90 degrees)

⇒ ∠1 = ∠4

Similarly, ∠2 = ∠3. Now, this gives

ΔQPM ~ ΔPRM (AA similarity)

PM/MR = QM/PM

⇒ PM² = QM × MR

5. ABC is an isosceles triangle right angled at C. Prove that AB² = 2AC².

Solution:

In △ABC, ∠ACB = 90°. We are given that △ABC is an isosceles triangle.

=> ∠A = ∠B = 45°

=> AC = BC

By Pythagoras theorem, we have:

AB² = AC² + BC²

= AC² + AC²  [∵ BC = AC]

= 2AC²

7.0Benefits of Class 10 Maths Chapter 6 Triangles 

1. Foundation for Advanced Geometry: It introduces essential geometric concepts that pave the way for higher-level studies in geometry and trigonometry.
2. Problem-Solving Skills: Students develop critical thinking by applying theorems like Pythagoras and congruence to solve real-world problems.
3. Real-Life Applications: Concepts of triangles are used in architecture, design, engineering, and navigation.
4. Logical and Critical Thinking: The chapter encourages logical reasoning and precision in problem-solving.
5. Competitive Exam Preparation: Triangles are a fundamental topic for exams like JEE and NEET.
6. Trigonometry Link: Understanding triangles is key to grasping trigonometric concepts.
7. Spatial Awareness: The chapter enhances the ability to visualize and manipulate geometric shapes.

In summary, the Triangles chapter builds strong mathematical foundations and practical skills for real-world applications and further studies.