Areas Related to Circles

12. Areas Related to Circles

Circle

The collection of points which is located at a certain distance from a certain point is said to be circle.

The certain distance is known as radius of the circle.

The certain point is known as centre of the circle.

The collection of points is known as circumference of the circle.

It is a geometrical figure in which infinite sides.

Area

The region which is surrounded by a closed geometrical figure is said to be area.

Calculation of area of a circle

When circle is divided into 16 equal parts from the centre of the circle. It is converted into a rectangle of length πr and breadth of r.

Important formulae of the circle

Area of cirle = πr²

Circumference of the circle = 2πr

Area of sector of a circle = πr²θ/360⁰

Area of minor segment = πr²2θ/360⁰ - r²sin2θ/2

Exercise 12.1

1. The radii of two circles are 19 cm and 9 cm respectively. Find the radius of the circle which has circumference equal to the sum of the circumferences of the two circles.

Solution:

Let the radius of the circle be R.

We know that

Circumference of the circle = C = 2πr

From question it is clear that

r₁ = 19 cm

r₂ = 9 cm

R = ?

Therefore,

C = C₁ + C₂

Therefore,

R = 2π(r₁ + r₂)/2π

Therefore,

R = 19 + 9

R = 28 cm Ans.

2. The radii of two circles are 8 cm and 6 cm respectively. Find the radius of the circle having area equal to the sum of the areas of the two circles.

Solution

Let the radius be r.

From question it is clear that

r =?

r₁ = 8 cm

r₂ = 6 cm

Therefore,

Radius of the circle = r² = π(r₁² + r₂²)/π

r² = 8² + 6²

r² = 64 + 36

r² = 100

r = 10 cm. Ans.

द्विघात समीकरण

द्विघात समीकरण को सामान्यतः

ax² + bx + c = 0

से निरूपित किया जाता है ,जहां a, b एवं c वास्तविक संख्या है तथा a का मान शून्य नही है।

द्विघात समीकरण में चर (x) का मान ज्ञात किया जाता है जिसे द्विघात समीकरण का मूल कहा जाता है।

किसी द्विघात समीकरण का दो और केवल दो ही मूल होता है जिसे क्रमशः α एवं β से निरूपित किया जाता है।

द्विघात समीकरण में a, b एवं c एक दूसरे से समन्धित होते है जो इस प्रकार है।

D = b² - 4ac

D को द्विघात समीकरण का विवेचक या विवित्तकर कहा जाता है।

D के मान से मूल की प्रकृति ज्ञात की जा सकती है।

1. मूल वास्तविक एवं आसमान होता है जब D का मान शून्य से बड़ा होता है।

2. मूल वास्तविक एवं समान होता है जब D का मान शून्य होता है

3. मूल वास्तविक नही होता है जब D का मान शून्य से छोटा होता है।

आगे जारी है ...........

9. Some Applications of Trigonometry

Class X

Exercise 9.1

NCERT Textbook Solution

Learning Points

• Angle of Elevation
• Angle of Depression

Angle of Elevation

The angle between line of sight and horizontal level when the observer is below the object is said to be angle of elevation.

Angle of Depression

The angle between line of sight and horizontal level when the observer is above the object is said to be the angle of depression.

Exercise 9.1

Q.No.1. 

 A circus artist is climbing a 20 m long rope, which is tightly stretched and tied from the top of a vertical pole to the ground. Find the height of the pole, if the angle made by the rope with the ground level is 30° (see Fig. 9.11).

Fig. 9.11

Solution:

From question it is clear that

Angle of Elevation = Ө = 30°

Height of the pole = AB = ?

Now applying trigonometric ratio sin𝞱

From figure it is clear that

sin𝞱 = AB/AC

Now putting the value of 

1/2 = AB/20

AB = 10 m Ans.

1. Real Numbers

Learning points in the point of view of 

Exercise 1.1 Class X Mathematics

• Euclid Division Lemma
• Euclid Division Algorithm
• Even Positive Integers
• Odd Positive Integers

Euclid Division Lemma

If a positive integer a is divided by b and remainder r is greater than or equal to zero but less than b. This statement is said to be Euclid Division Lemma.

In Mathematically it can be written as

a = bq + r

0≤ r⋖b

Euclid Division Algorithm

The stepwise calculation to find out the HCF of two positive integers by applying Euclid Division Lemma is said to be Euclid Division Algorithm.

Even Positive Integers

The positive integers which are completely divisible by 2 is said to be even positive integers.

Odd Positive Integers

The positive integers which are not completely divisible by 2 is said to be odd positive integers. 

Now we are able to solve the questions of Exercise 1.1

Introduction to Trigonometry .

Class X

Subject: Mathematics 

The ratios of sides in a right angle triangle is said to be trigonometric ratio.
The ratios of sides are as follows.
1. Perpendicular to hypotenuse
2. Base to hypotenuse
3. Perpendicular to base
4. Base to perpendicular
5. Hypotenuse to base
6. Hypotenuse to perpendicular

These ratios of sides are generally denoted by
1. sine (sin)
2. cosine (cos)
3. tangent (tan)
4. cotangent (cot)
5. secant (sec)
6. cosecant (cosec)

1.The opposite side of considering angle is identified as perpendicular.
2. The adjacent side of considering angle is identified as base.
3. The opposite side of right angle is identified as hypotenuse.

The right triangle ABC in which B is right angle. The considering angle can be either A or C.
When considering angle is C.
1.Perpendicular will be AB.
2. Base will be BC.
3. Hypotenuse will be AC.

When considering angle is A.
1. Perpendicular will be BC.
2. Base will be AB.
3. Hypotenuse will be AC.

There will be no change in the value of hypotenuse in both cases.

When considering angle is A.
The value of trigonometric ratios will be as follows.
1. sinA = BC/AC
2. cosA = AB/AC
3. tanA = BC/AB
4. cotA = AB/BC
5. secA = AC/AB
6. cosecA = AC/BC

When considering angle is C.
The value of trigonometric ratios will be as follows.

1. sinC = AB/AC
2. cosC = BC/AC
3. tanC = AB/BC
4. cotC = BC/AB
5. secA = AC/BC
6. cosecA = AC/AB

Here, the sum of angles A and C will be complementary or right angle.
Therefore,
A = 90⁰ - C
C = 90⁰ - A.
So,
sinA = cosC
cosA = sinC
tanA = cotC
cotA = tanC
secA = cosecC
cosecA = secC

We consider some special trigonometric ratios of 0⁰,30⁰, 45⁰, 60⁰ and 90⁰.

1.For 45⁰, we consider in an  isosceles triangle.
2. For 30⁰ and 60⁰ consider in an equilateral triangle.
3. For 0⁰ and 90⁰ coinsides the perpendicular and base with hypotenuse respectively.

All relations which are derived by Pythagoras theorem is said to be trigonometrical identities which is true for all values of considering angles.

In Right Angle Triangle ABC, in which angle B is right angle.

If considering angle is C which is θ.

AB² + BC² = AC²
When both sides divided by AC².

1. sin²θ + cos²θ = 1

This is a trigonometrical identity and true for all values of θ.

In this way, we find out rest other trigonometrical identities by dividing both sides by BC² and AC² respectively.

Rest others are as follows
4. 1 + tan²θ = sec²θ
5. sec²θ - tan²θ = 1
6. sec²θ - 1 = tan²θ
7. 1 + cot²θ = cosec²θ
8. cosec²θ - cot²θ = 1
9. cosec²θ - 1 = cot²θ

Others trigonometrical identities are derived from reciprocal of ratio of sides and complementary angle of rest angles of triangle.

As per reciprocal,
1. sinθ.cosecθ = 1
2. cosθ.secθ = 1
3. tanθ.cotθ = 1

As per complementary angle.

1. sinθ = cos(90⁰ - θ)
2. cosθ = sin(90 - θ)
3. tanθ = cot(90⁰ - θ)
4. cotθ = tan(90⁰ - θ)
5. secθ = cosec(90⁰ - θ)
6. cosecθ = sec(90⁰ - θ)

These are trigonometrical identities and true for each values of considering angle in right angle triangle.