Basic Geometrical Ideas | Speed Notes

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 = -x².

Convex Curves:

A curve is convex if it curves outward.

Examples:

   – The exterior of a circle.

   – The graph of a convex function like y = x².

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 180⁰ and less than 360⁰ (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 180⁰ and less than 360⁰ (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 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:

1. A ray parallel to the principal axis, which reflects through the focal point.
2. A ray passing through the focal point, which reflects parallel to the principal axis.
3. 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.