Circle

**Category : **4th Class

Circle is a close curved line whose all points are at the same distance from a given point in a plane.

**Centre of a Circle**

The point from which all the points of the curved line are at the same distance is called centre of the circle.

In the given figure, 0 is the centre of the circle.

**Radius of a Circle**

Distance between the centre and the curved line of a circle is called radius of the circle.

In the given figure, OA is the radius of the circle. Note: All the radius of a circle are equal in length.

**Chord of a Circle**

Any line segment which joins the two points of the curved line of a circle is called chord of the circle.

In the figure, AB is the chord of the circle.

**Diameter of a Circle**

The longest chord of a circle is called diametre of the circle. In other word the chord which passes through the centre is called diametre of the circle.

In the figure, AB is the diametre of the circle

**Note:** All the diametres of a circle are equal in length.

**Relation between Radius and Diametre of the Circle**

In a circle, diametre is twice of the radius. Let radius of a circle is r then diameter of the circle is 2r.

In the given circle, OM and ON are the two radius of the circle whereas MN is the diametre of the circle. We can see, MN = OM + ON ..........(i)

We know that all the radius of a circle is equal in length. Thus OM = ON ..........(ii)

So, we can write the equation (i) as followings: ' \[\text{MN}=\text{OM}+\text{OM or MN}=\text{ON}+\text{ON}\] Thus MN = 20M or MN = 2ON Thus, we see that the diametre of a circle is twice of the radius.

Radius of a circle is 5 cm. Find the diametre of the circle.

**Solution:** \[\text{Diametre}=\text{2}\times \text{radius}\] \[=\text{2}\times \text{5cm}\] =10 cm.

- Point indicates a particular location.
- Line segment is a fraction of line.
- Length of a line segment can be measured.
- Ray is considered as the extension of a line segment in one direction.
- Length of a ray cannot be measured.
- Line is considered as the extension of a line segment up to infinity in either direction.
- Angle is inclination between two rays which have common end point.
- Acute angle measures less than\[{{90}^{o}}\]whereas obtuse angle measures more than \[{{90}^{o}}\]
- Triangle is a close geometrical shape with three sides.
- Quadrilateral is a close geometrical figure with four sides.

- Point is a dimensionless mathematical object, it has no length, no width, no thickness.
- Sum of the length of any two sides of a triangle is always greater than that the length of the third side.

**Jack makes a line segment of 18 cm. He places an arrow at its one end. It is a** :

(a) Line

(b) Line segment

(c) Angle

(d) Ray

(e) None of these

**Answer (d)**

**Explanation-** Jacks places arrow at only one end of the line segment which means the line segment can be extended up to infinity in only one direction. Thus it is a ray.

**How many line segments are there in the figure?**

(a) 5

(b) 7

(c) 9

(d) 13

(e) None of these

**Answers (d)**

**Explanations-**

There are 13 line segments in the figure. \[\overline{\text{AC}},\text{ }\overline{\text{GC}},\text{ }\overline{\text{AB}},\text{ }\overline{\text{BC}},\text{ }\overline{\text{CD}},\text{ }\overline{\text{DE}},\text{ }\overline{\text{EB}},\text{ }\overline{\text{EC}},\text{ }\overline{\text{DG}},\text{ }\overline{\text{GF}},\]\[\overline{\text{FA}},\overline{F\text{E}},\]and\[\overline{\text{FC}}\].

**How many lines are there in the following figure?**

(a) 4

(b) 8

(c) 10

(d) 12

(e) None of these

**Answer (a)**

**How many rays are there in the following figure?**

(a) 11

(b) 8

(c) 12

(d) 15

(e) None of these

**Answer (d)**

**Which one of the following has no end points?**

(a) Line

(b) Line segment

(c) Ray

(d) All of these

(e) None of these

**Answer (a)**

**How many angles are there in the figure?**

(a) 7

(b) 11

(c) 9

(d) 4

(e) None of these

**Answer (b)**

**Explanation-** \[\angle \text{ABC},\text{ }\angle \text{BCA},\text{ }\angle \text{BAC},\text{ }\angle \text{APQ},\]\[\angle \text{AQP},\angle \text{BPQ},\text{ }\angle \text{CQP},\text{ }\angle \text{ARS},\text{ }\angle \text{ASR},\text{ }\angle \text{BRS},\]\[\text{and }\angle \text{CSR}\]

**Measure the angle given below and choose the correct option.**

(a) Acute angle

(b) Obtuse angle

(c) Right angle

(d) All of these

(e) None of these

**Answer (a)**

**Explanation-** Measurement of the given angle is\[\text{45}{}^\circ \]which is greater than\[0{}^\circ \]and smaller than\[\text{9}0{}^\circ \], thus it is an acute angle.

**Jack makes an angle which measures 67°. Which one of the following types of angle is this?**

(a) Obtuse angle

(b) Acute angle

(c) Right angles

(d) Straight angle

(e) None of these

**Answer (b)**

**Find the obtuse angle in the following triangle**

(a)\[\angle \text{ABC}\]

(b) \[\angle \text{BCA}\]

(c)\[\angle \text{BAC}\]

(d) All of these

(e) None of these

**Answer (a)**

**Name the angle which is formed between the minar and the base**.

(a) Acute angle

(b) Obtuse angle

(c) Right angle

(d) All of these

(e) None of these

**Answer (c)**

**Jack makes a triangle ABC, in which AB = AC. Which one of the following is true?**

(a)\[\angle \text{ABC}=\angle \text{BAC}\]

(b) \[\angle \text{ABC}=\angle \text{ACB}\]

(c)\[\angle \text{BAC}=\angle \text{ACB}\]

(d) All of these

(e) None of these

**Answer (b)**

**Explanation-** We know that opposite angles of equal sides in a isosceles triangle areequal. In triangle ABC, AB = AC, thus\[\angle \text{ABC}=\angle A\text{CB}\]. Therefore, option (b) is correct and rest of the options is incorrect.

**How many triangles are there in the figure?**

(a) 2

(b) 4

(c) 6

(d) 8

(e) None of these

**Answer (a)**

**Explanation -** There are two triangles in the above figure, \[\Delta \text{ABC}\] and triangle\[\Delta \text{DEF}\]. Therefore, options (a) is correct and rest of the options is incorrect.

**Look at the following triangle:**

** Which one of the following types of triangle is this?**

(a) Equilateral triangle

(b) Isosceles triangle

(c) Scalene triangle

(d) All of these

(e) None of these

**Answer (c)**

**PQR is a triangle. In which \[\angle \text{P}=\text{4}0{}^\circ \]and\[\angle \text{Q}=\text{27}{}^\circ \]. Triangle PRQ is an :**

(a) Acute angled triangle

(b) Right angled triangle

(c) Obtuse angled triangle

(d) All of these

(e) None of these

**Answer (c)**

**Look at the following picture: **

Which one of the following is correct?

(a) Both ZX and ZZ are acute angles.

(b) ZX is an acute angle whereas Z.7. is an obtuse angle.

(c) ZX is an obtuse angle whereas ZZ is an acute angle.

(d) Both ZX and ZZ are obtuse angles.

(e) None of these

**Answer (a)**

**Which one of the following is not true about a rectangle? **

(a) If all the angles of a quadrilateral are equal, the quadrilateral is a rectangle.

(b) A rectangle has four vertices.

(c) All sides of a rectangle are equal.

(d) Sum of all angles of a rectangle is\[\text{36}0{}^\circ \].

(e) None of these

**Answer (c)**

**Explanation -** All the sides of a rectangle may or may not be equal. The quadrilateral whose all angles are of\[\text{9}0{}^\circ \]but all the sides are not equal, is also a rectangle.

** Look at the following picture and choose the correct option :**

(a) If AB = BC, ABCD is a square

(b) If\[\text{AB}~\ne \text{BC}\], ABCD is a rectangle

(c) ABCD is a quadrilateral

(d) All of these

(e) None of these

**Answer (d)**

**In a rectangle all angles are:**

(a) Equal

(b) Unequal

(c) In different ratios

(d) Same ratios

(e) None of these

** Answer (a)**

**AB and AC are two radii of a circle. Which one of the following is not true.**

(a) Length of the diametre of the circle is equal to the sum of AB and AC.

(b) BC is the diametre of the circle.

(c) A is the centre of the circle.

(d) AB and AC are equal in length.

(e) None of these

**Answer (b)**

**Which one of the following is not true about a square?**

(a) Opposite sides of a square are equal.

(b) All the rectangles are also a square.

(c) A quadrilateral, whose all angles as well as sides are equal, is a square.

(d) All the angles of a square are right angles.

(e) None of these

**Answer (b)**

*play_arrow*Triangle*play_arrow*Angle*play_arrow*Quadrilateral*play_arrow*Circle*play_arrow*Introduction*play_arrow*Geometrical Figures*play_arrow*Geometrical Figures

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