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Did you study the Most Important Math Theorems as per NCERT Class 10 Syllabus?

Education

Did you study the Most Important Math Theorems as per NCERT Class 10 Syllabus?

Maths is all around us. Although academically considered a universal language, Maths is called the language of the universe and the queen or mother of all sciences. It is after all the secret to understanding the world. Being the language and logic of eve

By Topperlearning Expert 21st Jan, 2020 04:32 pm

Maths is a compulsory and core subject in school curriculum. It helps in the intellectual development of students. Students also study Math theorems which makes a student’s brain active.

On a personal level, Maths helps in a student’s intellectual, vocational, moral, spiritual, social and cultural development. However, Maths can get very difficult to understand, can be the hardest subject or may even get boring at times. So, we’ve analysed the types of questions in the CBSE Class 10 Maths syllabus and proof of theorems is a prominent type of question asked multiple times in the board exams. While these Math theorems may be tricky to understand at first, they form the most scoring part of the paper and so require to be practised during study and preparation.

So what is a Theorem?

A mathematical statement that can be proven is known as a theorem, and the process of showing a theorem to be correct is called a proof.

 

Here are the most important Math Theorems as per the NCERT Class 10 syllabus. You just have to understand the method of proving and practicing these theorems at the time of preparation for the exam.

 

I. Real Numbers:

Proof of irrationality of root (2)

Proof of irrationality of root (5)

 

II. Triangles:

1.  If a line is drawn parallel to one side of a triangle to intersect the other two sides at distinct points, the other two sides are divided in the same ratio.

Proof of basic proportionality theorem

2. The ratio of the areas of two similar triangles is equal to the ratio of the squares of their corresponding sides.

Proof of theorem on ratio of the areas of two similar triangles

3. In a right-angled triangle, the square of the hypotenuse is equal to the sum of the squares of the other two sides.

Proof of Pythagoras’ theorem

4. In a triangle, if the square of one side is equal to the sum of the squares of the other two sides, then the angles opposite to the first side is a right angle.

Proof of converse of Pythagoras’ theorem

 

III. Introduction to Trigonometry

Prove that the trigonometric identity sin2A + cos2A = 1.

Proof of sin2x + cos2x = 1.

 

IV. Circles:

Tangents to a circle motivated by chords drawn from points coming closer and closer to the point.

1. Prove that the tangent at any point of a circle is perpendicular to the radius through the point of contact.

Proof: Angle between a tangent and the radius of the same circle is 90 degrees.

2. Prove that the lengths of tangents drawn from an external point to a circle are equal.

Proof: Tangents drawn from an external point to a circle are equal.

Now let’s see an important theorem from the CBSE Class 10 syllabus and try to understand how to proceed with its proof.

Pythagoras Theorem:

In a right-angled triangle, the square of the hypotenuse is equal to the sum of the squares of the other two sides.

With the help of the given statement, we will first draw a diagram of a right-angled triangle.

 

Next we will write down

  1. What is given
  2. What is to be proven
  3. Construction (if required)
  4. Method of proof

Given: ∆ABC is a right-angled triangle, right angled at B.

To prove: AC2 = AB2 + BC2

Construction: Draw BD ^ AC

 

Proof:

∆ADB  ~ ∆ABC              … (If a perpendicular is drawn from the vertex of the right angle of a right triangle to the hypotenuse, then triangles on both sides of the perpendicular are similar to the whole triangle and to each other.)

⇒    ….. (Sides are proportional)

⇒    AB2 = AD × AC           … (1)

∆ADB  ~ ∆ABC              … (If a perpendicular is drawn from the vertex of the right angle of a right triangle to the hypotenuse, then triangles on both sides of the perpendicular are similar to the whole triangle and to each other.)

⇒    ….. (Sides are proportional)

⇒   BC2 = CD × AC           … (2)

Adding equations (1) and (2), we get

AB2 + BC2 = AD × AC + CD × AC 

                  = AC (AD + CD)

                  = AC × AC

                  = AC2  

⇒ AB2 + BC2 = AC2  

Hence, it is proved that in a right-angled triangle, the square of the hypotenuse is equal to the sum of the squares of the other two sides.

By following the above method, you will surely score full marks for this theorem.

 

Do’s and Don’ts while preparing for Math theorems:

Do’s

Don’ts

1. Draw a neat and clean diagram with a pencil.

1. By heart a proof

2. These steps should be included:

  • Statement
  • Given
  • To prove
  • Proof
  •  Construction (if required)

2. Skip steps while proving the theorem

3. Conclude the result properly

3. Not writing a proper reason for a step 

 

Best practices for remembering Math theorems: 

1. Make a separate notebook for theorems

2. Understand each theorem thoroughly

3. Ask questions in class if you don’t understand any step of the proof

4. Understand and by heart the statements of theorems

5. Practise theorems on a daily basis

6. Prepare flow charts

7. Try to explain the proof to your friends

8. Provide enough time for preparation of theorems

9. Don’t leave any theorem for the last minute

10. Take your own test, i.e. test yourself

Drawbacks of memorizing Math theorems:

1. You can forget them completely

2. Doesn’t allow for a deeper understanding of a theorem

3. No connection between old and new knowledge

4. You may lose focus

5. Won’t be able to develop problem-solving techniques

So, this was all about the most important Math theorems as per the NCERT syllabus for Class 10. Keep practising, never stop trying and don’t get stressed. Push yourself because no one else is going to do it for you.

Just believe in yourself. And if you need any help, you can ask us your queries at Ask the Expert—Your love for learning Maths is bound to grow exponentially!

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