## Pre-Requisires

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**Basic Geometrical Ideas | Speed Notes**

**Notes For Quick Recap**

**Geometry**

**Geometry** is a branch of mathematics concerned with properties of space such as the distance, shape, size, and relative position of figures. A mathematician who works in the field of geometry is called a 🎉**geometer**. **(Scroll down to continue …)**

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**Basic Geometrical Ideas | Speed Notes**

**Notes For Quick Recap**

**Geometry**

**Geometry** is a branch of mathematics concerned with properties of space such as the distance, shape, size, and relative position of figures. A mathematician who works in the field of geometry is called a 🎉**geometer**.

### Space

**Space:** space refers to a set of points that form a particular type of structure.

### Plane

A plane is a flat surface that extends infinitely in all directions

### Point

Point is an exact position or location in space with no dimensions.

Mathematically, a point is defined as a circle with zero radius.

Since it is not possible its is represented by a very small dot.

A point is usually represented by a capital letter.

In mathematical terms, pont is a cirlce with no radius. It does mean that a very very small circle.

A point determines a location. It is usually denoted by a capital letter.

## Lines And Its Types

### Ray

A Ray is a straight path that stars at a point and extends infinitely in one direction.

**Note: **A ray is a portion of line starting at a point and extends in one direction endlessly. A ray has only one endpoint (Initial point).

### Line or Straight Line

A line is a straight path that extends infinitely in two opposite directions. It can be treated as a combination of two rays starting from the same point but extending in the opposite directions.

**Note: **A line has no end points.

### Line Segment:

A line segment is the part of a line between two points. (Segment means part).

The length of a line segment is the shortest length between two end points.

The line segment has two end points. **Note: **A line Segment has two endpoints. (both Initial and end points). **(Scroll down to continue …)**.

**Intersecting Lines and Non-intersecting Lines**

## Intersecting And Parallel Lines

### Parallel Lines

The lines which never cross each other at any point are called **Parallel lines** or** Non-intersecting lines**.

In other words, lines that are always the same distance apart from each other and that never meet are called **Parallel lines Or Non-intersecting lines.**

The perpendicular length between two lines is the **distance between parallel lines**.

**Note:** Parallel lines do not have any common point.

### Intersecting Lines

The lines which cross (meet) each other at a point are called Intersecting Lines or non-parallel lines.

Intersecting lines meet at only one point.

## Angle

An **angle **is made up of two rays starting from a **common end point**.

An angle leads to three divisions of a region:

**On the angle**, the **interior of the angle **and the **exterior of the angle**.

## Curve

In geometry, a curve is a line or shape that is drawn smoothly and continuously in a plane with bends or turns.

** **In other words, Curve is a drawing (straight or non-straight) made without lifting the pencil may be called a **curve**.

Mathematicians define a curve as any shape that can be drawn without lifting the pen.

In Mathematics, A curve is a continuous and smooth line that is defined by a mathematical function or parametric equations.

**Note:** In this sense, **a line is also a curve.**

**Types of Curves**

**Simple Or Open And Closed Curves**

Cureves are two types based on intersection (crossing). They are (i) Simple or Open curve (ii) Closed curve.

**Simple Curve**

Simple or open curve is a curve that does not cross (intersect) itself.

**Closed Curve**

Closed curve is a curve that crosses (intersects) itself.

**Concave And Convex Curves**

Curves are of two types. They are concave curve and convex curve.

**Concave Curve:**

A curve is concave is a curve that curves inward, resembling a cave.

**Examples**:

– The interior of a circle.

– The graph of a concave function like y = -x^{2}.

**Convex Curves:**

A curve is convex if it curves outward.

**Examples:**

– The exterior of a circle.

– The graph of a convex function like y = x^{2}.

## Polygon

A polygon is a simple closed figure formed by the line segments.

**Types of Polygons**

Polygons are classified into two types on the basis of interior angles: as (i) Convex polygon (ii) Concave polygon.

**(a) Concave Polygon:**

A concave polygon is a simple polygon that has at least one interior angle, that is greater than 180^{0} and less than 360^{0 }(Reflex angle).

And at Least one diagonal lies outside of the closed figure.

Atlest one diagonal lies outside of the polygon.

**b. Convex Polygon: **

A convex polygon is a simple polygon that has at least no interior angle that is greater than 180^{0} and less than 360^{0 }(Reflex angle).

And no diagonal lies outside of the closed figure.

In this case, the angles are either acute or obtuse (angle < 180 ^{o}).

**Regular And Irregular Polygon**

On the basis of sides, there are two types of polygons as **Regular Polygon** and **Irregular Polygon**

(**a) Regular Polygon:**

A convex polygon is called a regular polygon, if all its sides and angles are equal as shown in the following figures.

Each angle of a regular polygon of n-sides =

#### Part of Polygon

**(i) Sides Of The Polygon**

** **The line segments of a polygone are called sides of the polygon.

**(ii) Adjacent Sides Of Polygon**

**Adjacent sides** of a polygon are thesides of a polygon with a common end point.

**(iii) Vertex Of Polygon**

**Vertext of a polygon** is a point at which a pair of sides meet.

**(iv) Adjacent Vertices**

Adjacent vertices of polygon are the end points of the same side of the polygon.

**(v) ****diagonal**

**diagonal**

** Diagonal** of a polygone is a line segment that joins the non-adjacent vertices of the polygon.

## Triangle

A **triangle** is a three-sided polygon.

In other terms triangle is a three sided closed figure.

## Quadrilateral

A **quadrilateral **is a four-sided polygon. (It shouldbe named cyclically).

In any

similar relations exist for the other three angles.

## Circle And Its Parts

A **circle** is the path of a point moving at the same distance from a fixed point.

### Centre Of Circle

**Centre Of Circle** is a point that is equidistant from any point on the boundary of the circle.

In other words the centre of the circles is a centre point of the circle.

### Radius Of Circle

Radius of circle is the distance between the centre of the circle and any point on the boundary of the circle.

### Circumference Of Circle

Circumference of circle is the length of the boundary of the circle.

### Chord Of Circle

**A chord **of a circle is a line segment joining any two points on the circle.

A **diameter** is a chord passing throughthe Centre of the circle.

### Sector Of Circle

A sector is the region in the interior of a circle enclosed by an arc on one side and a pair of radii on the other two sides.

### Segment Of Circle

A segment of a circle is a region in the interior of the circle enclosed by an arc and a chord.

The diameter of a circle divides it into two semi-circles.

The diameter of a circle divides it into two semi-circles.

### 1. Find the distance of the image when an object is placed on the principal axis at a distance of 10 cm in front of a concave mirror whose radius of curvature is 8 cm. (AS1)

**Answer:**

Given:

- Object distance, ( u = -10 ) cm (negative because the object is in front of the mirror)
- Radius of curvature, ( R = 8 ) cm

The focal length, ( f ), is given by: [ f = \frac{R}{2} = \frac{8}{2} = 4 \text{ cm} ]

Using the mirror formula: $$ \frac{1}{f} = \frac{1}{v} + \frac{1}{u} $$

Substitute the known values: $$ \frac{1}{4} = \frac{1}{v} + \frac{1}{-10} $$

$$ \frac{1}{v} = \frac{1}{4} + \frac{1}{10} $$

$$ \frac{1}{v} = \frac{5 + 2}{20} = \frac{7}{20} $$

$$ v = \frac{20}{7} \approx 2.86 \text{ cm} $$

So, the image distance is approximately ( 2.86 ) cm in front of the mirror.

### 2. The magnification produced by a mirror is +1. What does it mean? (AS1)

**Answer:**

A magnification of ( +1 ) means that the image formed by the mirror is:

- The same size as the object.
- Erect (upright).
- Virtual (since real images formed by concave mirrors are inverted).

### 3. If the spherical mirrors were not known to human beings, guess the consequences. (AS2)

**Answer:**

Without spherical mirrors, several technological and practical applications would be impacted:

**Automotive Safety:**Rear-view and side mirrors in vehicles would be less effective, reducing driver visibility and increasing accident risks.**Optical Instruments:**Devices like telescopes, microscopes, and cameras would be less efficient or non-existent, hindering scientific progress.**Daily Use:**Personal grooming mirrors would be less effective, affecting daily routines.

### 4. Draw suitable rays by which we can guess the position of the image formed by a concave mirror? (AS5)

**Answer:**

To determine the image position, you can draw the following rays:

- A ray parallel to the principal axis, which reflects through the focal point.
- A ray passing through the focal point, which reflects parallel to the principal axis.
- A ray passing through the center of curvature, which reflects back on itself.

### 5. Show the formation of image with a ray diagram when an object is placed on the principal axis of a concave mirror away from the center of curvature? (AS5)

**Answer:**

Here’s a ray diagram for an object placed beyond the center of curvature ©:

!Concave Mirror Ray Diagram

In this case:

- The image is formed between the focal point (F) and the center of curvature ©.
- The image is real, inverted, and smaller than the object.

### 6. Why do we prefer a convex mirror as a rear-view mirror in vehicles? (AS7)

**Answer:**

Convex mirrors are preferred for rear-view mirrors in vehicles because:

They produce smaller, upright images, which help in better judgment of distances and speeds of approaching vehicles.!

They provide a wider field of view, allowing drivers to see more area behind them.

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- WHOLE NUMBERS | Study
- KNOWING OUR NUMBERS | Study
- PLAYING WITH NUMBERS | Study
- BASIC GEOMETRICAL IDEAS | Study
- UNDERSTANDING ELEMENTARY SHAPES | Study
- INTEGERS | Study
- FRACTIONS | Study
- DECIMALS | Study
- DATA HANDLING | Study
- MENSURATION | Study
- ALGEBRA | Study
- RATIO AND PROPORTION | Study
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