CBSE Class 10

On this page, you will find Constructions Class 10 Notes Maths Chapter 11 Pdf free download. CBSE NCERT Class 10 Maths Notes Chapter 11 Constructions will seemingly help them to revise the important concepts in less time.

CBSE Class 10 Maths Chapter 11 Notes Constructions

Constructions Class 10 Notes Understanding the Lesson

Division of a line segment internally in the given ratio.

Let AB be a line segment of certain length. We need a point P on AB dividing it internally in the ratio m : n
Steps of Construction: Let m = 4, n = 3.

• Draw a line segment AB of given length.
• Make an acute ∠BAX with AB
• Use a compass of any radius and mark 7(i.e. m + n) points A₁, A₂,…… , A₇
  such that AA₁ = A₁A₂ = A₂A₃ =………………= A₆A₇
  
  
• Join BA₇
• Through the point A₄ [m = 4], draw a line parallel to BA₇ by making an angle equal to AA₇B at A₄ intersecting AB at P. Then AP : PB = 4 : 3.

Construction of a Triangle similar to a given Triangle as per given scale factor.
1. Scale Factor  \(\frac{m}{n}\)(where m < n)
Steps of construction: Let \(\frac{m}{n}=\frac{3}{4}\)

• Construct a triangle ABC by using given data.
• Make an acute angle ∠BAX below the base AB.
• Along AX, mark 4 points [the greater of 3 and 4 in \(\frac{3}{4}\)] as A₁, A₂,A₃, A₄ such that
  AA₁ = A₁A₂ = A₂A₃ = A₃A₄.
• Join A₄ B
  
• From A₃, draw A_gB’ || A₄B, meeting AB at B’.
  From B’, draw B’C’ || BC, meeting AC at C’.

Thus, ΔAB’C’ is the required triangle each of whose sides is \(\frac{3}{4} \)of the corresponding side of ΔABC.

2. Scale Factor \(\frac{m}{n} \) (where m > n)
Steps of construction:
Let \(\frac{m}{n}=\frac{5}{3}\)

• Construct ΔABC using given data.
• Make an acute angle ∠BAX below the base AB. Extend AB to AY and AC to AZ.

• Along AX, mark 5 points [the greater of 5 and 3 in \(\frac{5}{3} \)
  such that AA₁ = A₁A₂ =………. = A₄A₅.
• Join A₃B.
• From A_g, draw A₅B’ || A₃B, meeting AY produced at B’.
• From B’, draw B’C’ || BC, meeting AZ produced at C’.
  Thus, ΔAB’C’ is the required triangle, each of whose sides is \(\frac{5}{3}\) of the corresponding side of ΔABC.

Construction of the pair of tangents from an external point to a circle.

Let O is the centre of the circle and a point A is external point to a circle.

Steps of construction

• Join AO and bisect it. Let M be the mid-point of AO.
• Taking M as centre and MO as radius, draw a circle.
  Let it intersects the given circle at the points B and C.
• Join AB and AC.
  Thus, AB and AC are the required tangents.

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CBSE Class 10 Maths Chapter 10 Notes Circles

Circles Class 10 Notes Understanding the Lesson

1. Circle: A circle is a collection of all points in a plane which are at a constant distance from a fixed point. Here
Fixed point is called centre of the circle. Constant distance is called radius of the circle.

2. Considering a circle with centre ‘O’ and radius V and a line T in a plane.
Three different situations are there:

• When there is no common point between the circle and the line. Then the line is known as a non¬intersecting line
• When the line passes the circle in two points, the line is called a secant
• When a line meets the circle at a point, the line is called a tangent
  The point at which the tangent touches the circle is called point of contact.

Facts related to tangent of circle.

Given: A circle with centre 0 and radius r.
A tangent XY at point P to the circle.
To prove: OP ⊥ XY
Construction: Take a point Q on XY other than P. Join OQ.
Proof: Point Q lies outside the circle.

If point Q lies inside the circle then XY will become a secant and not a tangent to the circle.
∴ OQ > OP
which is true for every point on the line XY except the point P.
⇒ OP is the shortest of all the distances of the point O to the points of XY.
OP ⊥ XY
[ ∵ Shortest length from the point outside the line to the line is perpendicular]
Remark: A line drawn through the end point of radius and perpendicular to it, is the tangent to the circle.

Theorem 10.2: The lengths of tangents drawn from an external point to a circle are equal.
Given: PT and PS are tangents from point P to circle with centre 0.
To prove: PT = PS
Construction: Join OP, OT and OS.
Proof: In ΔOTP and ΔOSP
OT = OS [Radii of same circle]

OP = QP
∠OTP = ∠OSP [Each 90º]
∴ ΔOTP ≅ ΔOSP  [RHS]
⇒ PT = PS   [CPCT]

On this page, you will find Some Applications of Trigonometry Class 10 Notes Maths Chapter 9 Pdf free download. CBSE NCERT Class 10 Maths Notes Chapter 9 Some Applications of Trigonometry will seemingly help them to revise the important concepts in less time.

CBSE Class 10 Maths Chapter 9 Notes Some Applications of Trigonometry

Some Applications of Trigonometry Class 10 Notes Understanding the Lesson

Trigonometry is the study of relationships between the sides and angles of a triangle. In this chapter you will study about some ways in which trigonometry is used:

• It is used in geography and in navigation.
• It is used in constructing maps, determine the position of an island in relation to the longitudes and latitudes.
• It is used for calculating the height and distance of various objects without measuring it.

Terms related to height and distance:

1. Line of sight: The line joining the eyes of the observer and the objects which he/she observes is called line of sight.

2. Angle of elevation: (When object is above the horizontal)
The angle between the line of sight and the horizontal is called the angle of elevation.

3. Angle of depression: (When object is below the horizontal)
The angle between the horizontal line and the line of sight is called the angle of depression.

Trigonometric formulae used:

On this page, you will find Introduction to Trigonometry Class 10 Notes Maths Chapter 8 Pdf free download. CBSE NCERT Class 10 Maths Notes Chapter 8 Introduction to Trigonometry will seemingly help them to revise the important concepts in less time.

CBSE Class 10 Maths Chapter 8 Notes Introduction to Trigonometry

Introduction to Trigonometry Class 10 Notes Understanding the Lesson

The word trigonometry is derived from the Greek words ‘Tri’ which means three, ‘gon’ means sides and metron meaning measure.

It means trigonometry is the study of relationship between the sides and angles

• The earliest work on trigonometry was recorded in Egypt and Babylon.
• Trigonometry was used by early astronomers to find out the distance of stars and planets from the earth.

Trigonometric Ratios

The ratios of the sides of a right triangle with respect to its acute angles are called trigonometric ratios.

1. In right triangle, side opposite to given acute angle will always be perpendicular.
2. Side opposite 90° will always be hypotenuse.
3. Remaining side will be base.
4. The sum of two angles (except right angle) is 90°
   i.e.,      ∠A + ∠C = 90°           ( ∵ ∠B = 90°)
   

1. sin θ = sin θ = \(\frac{0}{\mathrm{H}}\) (O- side opposite to given angle i.e., acute angle)

2. cosine θ = cos θ = \(\frac{\mathrm{A}}{\mathrm{H}} \)(A-adjacent side) (H-Hypotenuse)

3. Tangent θ = tan θ =\(\frac{\mathrm{O}}{\mathrm{A}}\)
(O-side opposite to acute angle, A-adjacent side)

4. cosecant θ = cosec θ= \(\frac{\mathrm{H}}{\mathrm{O}}\)

5. secant θ = sec θ =\(\frac{\mathrm{H}}{\mathrm{A}}\)

6. cotangent θ= cot θ =\(\frac{\mathrm{A}}{\mathrm{O}}\)

Trigonometric angles for some specific angles

Also we can find values for some special angles as follows:

Trigonometric Identities

Identity: That equation is called an identity. If it is true for all values of the variables which involved. I. In right ΔABC,

On this page, you will find Coordinate Geometry Class 10 Notes Maths Chapter 7 Pdf free download. CBSE NCERT Class 10 Maths Notes Chapter 7 Coordinate Geometry will seemingly help them to revise the important concepts in less time.

CBSE Class 10 Maths Chapter 7 Notes Coordinate Geometry

Coordinate Geometry Class 10 Notes Understanding the Lesson

Distance formula

1. The distance between two points A(x₁, y₁) and B(x₂, y₂) is

2. The distance of point A(x, y) from origin 0(0, 0) is
\(\mathrm{AO}=\sqrt{x^{2}+y^{2}}\)

3. Three given points will form:

• Right angled triangle if sum of squares of any two sides is equal to the square of third (largest) side.
• Equilateral triangle if length of all three sides are equal.
• Isosceles triangle if length of any two sides are equal.
• A line or collinear if sum of two sides is equal to third side.

4. Four given points will form:

• Square if length of all four sides are equal and diagonals are equal.
• Rhombus if length of all four sides are equal.
• Rectangle if opposite sides are equal and diagonals are equal.
• Parallelogram if opposite sides are equal.

Section formula
I. If A(x₁, y₁) and BB(x₂, y₂)) are two points on a plane and P(x, y) divides AB internally in the ratio m : n, then co-ordinates of P are given by

Area of a Triangle

1. Area of ΔABC formed by vertices A(x₁, y₁), B(x₂, y₂) and C(x₃, y₃) is given by
Ar(ΔABC) =\(\frac{1}{2}\) (- y₃) + x₂ (y₃ – y₁)+ x₃(y₁ – y₂)]
[Only positive numerical value to be taken]

2. If Ar(ΔABC) = 0, then A, B and C are collinear points.

3. If‘C’ is centroid of a triangle, the median is divided in the ratio 2 : 1 by C and coordinates of C are
\(\left(\frac{x_{1}+x_{2}+x_{3}}{3}, \frac{y_{1}+y_{2}+y_{3}}{3}\right)\)

4. Area of quadrilateral LMNO = ar(ΔLMO) + ar(ΔNMO)