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The word polygon takes its origin from the Greek, poly, meaning many and gon, meaning angle. Polygons are 2 dimensional closed shapes made up of straight lines. Sometimes the interior of a polygon is known as its body.

We see polygons all around us. The school building is having a square or rectangular walls. You may come across some houses with triangular roofs. The tiles laid on the floor may be in square or hexagonal shape. 

In this chapter, you will learn about polygons and their classifications.

Polygon: It is a closed figure bounded by straight line segments.

Some important facts about polygon:

• Line segments forming a polygon are called sides of the polygon.
• The point where two sides of a polygon meet are called the vertex of the polygon.
• The line segment containing two non-adjacent vertices is called the diagonal of the polygon.
• The angle formed at the vertices inside the closed figure are called interior angles.

Classification of polygons:

Polygons are classified according to the number of sides or vertices they have in to:

• Traingle
• Quadrilateral
• Pentagon
• Hexagon
• Heptagon
• Octagon
• Nonagon
• Decagon 

The figure below on the right side shows these polygons.

CONCAVE & CONVEX POLYGONS:
We know that each side of a polygon is connected by two consecutive vertices of the polygon.
A diagonal is a line segment that connects the non-consecutive vertices of a polygon. 
If a diagonal lies outside a polygon, then the polygon is called a concave polygon.
If all the diagonals lie inside the polygon, then the polygon is said to be a convex polygon.

A polygon with all its sides equal and all its interior angles equal is said to be a regular polygon. In this chapter, you will learn about regular polygons.

REGULAR & IRREGULAR POLYGONS:

A regular polygon is equiangular and equilateral. The word equiangular means, the interior angles of the polygon are equal to one another. The word equilateral means, the lengths of the sides are equal to one another.
The polygon with unequal sides and unequal angles is called an irregular polygon.

ANGLE SUM PROPERTY:
The sum of all interior angles of a polygon is called the angle sum.
.           
At one vertex, we extend a side. This side makes an angle with its consecutive side. This angle is called the exterior angle. The interior angle and the exterior angles are adjacent angles.  These angles form a linear pair. Hence the sum of the exterior angles of any polygon is 360°.

Let us consider some examples:

Example 1:

Find the sum of all the interior angles of a polygon having 29 sides.

Solution:

We know that sum of all the interior angle in a polygon = (n – 2) × 180°.

Here, n = 29.

Therefore, the sum of all interior angles = (29 – 2) × 180°.

                                                                = 27 × 180°.

                                                                = 4860°.

Example 2: Is it possible to have a polygon, the sum of whose interior angle is 9 right angles?

Solution:

Let the number of sides be n.

The sum of all interior angles = (2n – 4) × 90°.

So, (2n – 4) × 90° = 9 × 90°.

n = 6.5, hence it is not possible to have a polygon the sum of whose interior angles is 9 right angles.

Example 3: The sides of a Pentagon are produced in order. If the measure of exterior angles so obtained are x°,2 x°,3 x°,4 x°,5 x° and so on, find all exterior angles

Solution:

The sum of exterior angles = 360°.

So, x° + 2x° + 3x° + 4x° + 5x° = 360°.

15x° = 360°.

x° = 24°.

Hence, the exterior angles are 24°,  48°,  72°,  96°, and  120°.

Example 4: If each interior angle of a regular polygon is 144°, Find the number of sides in it.

Solution:

Let the number of sides be n.

Each interior angle = ((2n – 4) × 90°) / n.

144 = 180n – 360°.

n = 10.

The word quadrilateral is the combination of two Latin words quadri, meaning four, and latus, meaning side. You come across quadrilaterals every day. For example, the page of a book, the top of a pencil box, the top of a dining table and so on, are all quadrilaterals (rectangular shape). 

A simple closed figure formed by joining four line segments is called a quadrilateral.  It has four sides, four angles, four vertices and two diagonals.

In this chapter, you will learn about basic properties of a quadrilateral.

In a quadrilateral ABCD :

• The four points A, B, C, D are called its vertices.
• The four line segments AB, BC, CD and DA are called its sides.
• ∠DAB, ∠ABC, ∠BCD and ∠CDA are called its angles, to be denoted by ∠A, ∠B, ∠C and ∠D respectively.
• The line segments AC and BD are called its diagonals.

Some important facts about quadrilateral:

• If each angle of a quadrilateral is less than 180°, then it is called convex quadrilateral.
• If each angle of a quadrilateral is greater than 180°, then it is called concave quadrilateral.

Angle Sum Property of quadrilateral:

• Sum of interior angles of quadrilateral is 180°

Let us consider some examples:

Example 1:

The three angles of a quadrilateral are 76°, 54° and 108°. Find the measure fourth angle.

Solution:

We know that sum of the angles of a quadrilateral is 360°.

Let the unknown angle be x

76°+  54° +  108° + x = 360°.

 x =  122°.

Example 2: The angles of a quadrilateral are in the ratio of 3 : 4 : 5 : 6.Find all its angles.

Solution:

Let the angles be 3x°, 4x°, 5x° and 6x°.

3x° + 4x° + 5x° + 6x° = 360°.

18x° = 360°.

x° = 20°.

Hence, the angles are 60°, 80°, 100° and 120°.

Example 3: The angles of a quadrilateral are in the ratio of 4 : 6 : 3. If the fourth angle is 100°, find the other angles of a quadrilateral.

Solution:

Let the angles be 4x°, 6x° and 3x°.

4x° + 6x° + 3x° + 100° = 360°.

13x° = 260°.

x° = 20°.

Hence, the angles are  80°,120° and  60°,

TYPES OF QUADRILATERAL

A closed figure with four sides is a quadrilateral. We come across many different types of quadrilaterals every day. It would be interesting to know the types of quadrilaterals, their shapes and basic properties.

Parallelogram:

A quadrilateral is called a parallelogram, if both pairs of its opposite sides are parallel. 

In the figure given below, ABCD is a quadrilateral in which:

AB ∥ DC and AD ∥ BC. 

So, ABCD is a parallelogram.

Rhombus:

A parallelogram having all sides equal is called a rhombus.

In the figure given below, ABCD is a rhombus in which:

AB ∥ DC, AD ∥ BC and AB = BC = CD = DA.

Rectangle:

A parallelogram in which each angle is a right angle is called a rectangle.

In the figure given below, ABCD is a quadrilateral in which:

AB ∥ DC, AD ∥ BC and ∠A = ∠B = ∠C = ∠D = 90°.

So, ABCD is a rectangle.

Square:

A parallelogram in which all the sides are equal and each angle measures 90° is called a square.

In the figure given below, ABCD is a quadrilateral in which:

AB ∥ DC, AD ∥ BC, AB = BC = CD = DA.

and ∠A = ∠B = ∠ C = ∠D = 90°.

So, ABCD is a square.

Trapezium:

A quadrilateral having exactly one pair of parallel sides is called a trapezium.

In the figure given below, ABCD is a quadrilateral in which AB ∥ DC. So, ABCD is a trapezium.

If non–parallel sides of a trapezium are equal, it is called an isosceles trapezium.

Kite:

A quadrilateral is called a kite if it has two pairs of equal adjacent sides but unequal opposite sides.

In the figure given below, ABCD is a quadrilateral with AB = AD, BC = DC, AD ≠ BC and AB ≠ DC.

So, ABCD is a kite.

Let us consider an example:

Example: In the square PQRS given in the figure below, PQ = 3x – 7 and  QR= x + 3 , find PS.

Solution:

As all sides are equal so, PQ = QR.

3x – 7 = x + 3.

2x =  10.

 x = 5.

PQ = 3x – 7 = 8.

QR = x + 3 = 8.

Hence PS = 8.

Construction of geometrical objects is an important part of geometry. This is nothing but practical geometry. Every object is constructed using certain instruments following certain rules and methods. In this chapter, you will learn about how to construct line segments, angles, triangles and circles by using tools such as ruler and compass.

Things To Remember

• Constructing a circle given its radius:
  1. Open the compasses and measure the required radius, say 3 cm.
  2. Mark a point O with a sharp pencil where we want the centre of the circle to be.
  3. Place the pointer of the compasses on O.

• Rotate the compasses to draw the circle. Take care to complete the movement around in one instant.

• Constructing a line segment of given length:
  1. Draw a line l and mark a point A on it.
  2. Place the compasses pointer on zero mark of the ruler. Open it to place the pencil point upto the required length.
  3. Taking caution that the opening of the compasses has not changed, place the pointer of the compasses at A and make an arc to cut l at B.
  4. AB is the line segment of the required length.
• Constructing a copy of a given line segment:
  1. Given \ov{AB} whose length is not known.
  2. Fix the compasses pointer on A and the pencil end on B. The opening of the instrument now gives the length of \ov {AB}.
  3. Draw any line l. Choose a point C on l. Without changing the compasses setting, place the pointer on C.
  4. Swing an arc that cuts l at a point, say, D. Now \ov{CD} is a copy of \ov{AB}.
• Constructing perpendicular to a line through a point on it:
  1. Given a point P on a line l.
  2. With P as the centre and a convenient radius, construct an arc intersecting the line l at two points A and B.
  3. With A and B as centres and a radius greater than AP construct two arcs, which cut each other at Q.
  4. Join PQ. Then PQ is perpendicular to l. We write {PQ}↖{↔} ⊥ l.
• Constructing perpendicular to a line through a point not on it:
  1. Given a line l and a point P not on it.
  2. With P as the centre, draw an arc which intersects line l at two points A and B.
  3. Using the same radius and with A and B as centres, construct two arcs that intersect at a point, say Q, on the other side.
  4. Join PQ. Then PQ is perpendicular to l. We write {PQ}↖{↔} ⊥ l.
• Constructing the perpendicular bisector of a line segment:
  1. Draw a line segment \ov{AB} of any length.
  2. With A as the centre, using compasses, draw a circle. The radius of your circle should be more than half the length of \ov{AB}.
  3. With the same radius and with B as the centre, draw another circle using compasses. Let it cut the previous circle at C and D.
  4. Join \ov{CD}. It cuts \ov{AB} at O. Use your divider to verify that O is the midpoint of \ov{AB}. Also, verify that ∠COA and ∠COB are right angles. Therefore, \ov{CD} is the perpendicular bisector of \ov{AB}.
• Constructing an angle of a given measure:
  1. Draw \ov{AB} of any length.
  2. Place the centre of the protractor at A and the zero edge along \ov{AB}.
  3. Start with zero near B. Mark point C at the required angle measure, say 40°.
  4. Join AC. ∠BAC is the required angle.
• Constructing a copy of an angle of unknown measure:
  Given ∠A, whose measure is not known.
  1. Draw a line l and choose a point P on it.
  2. Place the compasses at A and draw an arc to cut the rays of ∠A at B and C.
  3. Use the same compasses setting to draw an arc with P as the centre, cutting l in Q.
  4. Set your compasses to the length BC with the same radius.
  5. Place the compasses pointer at Q and draw the arc to cut the arc drawn earlier in R.
  6. Join PR. This gives us ∠P. It has the same measure as ∠A. This means ∠QPR has the same measure as ∠BAC.

• Constructing bisector of an angle:
  Let an angle, say, ∠A be given.
  1. With A as the centre and using compasses, draw an arc that cuts both rays of ∠A. Label the points of intersection as B and C.
  2. With B as the centre, draw (in the interior of ∠A) an arc whose radius is more than half the length BC.
  3. With the same radius and with C as the centre, draw another arc in the interior of ∠A. Let the two arcs intersect at D. Then AD is the required bisector of ∠A.
• Constructing a 60° angle:
  1. Draw a line l and mark a point O on it.
  2. Place the pointer of the compasses at O and draw an arc of convenient radius which cuts the line at a point say, A.
  3. With the pointer at A (as centre), now draw an arc that passes through O.
  4. Let the two arcs intersect at B. Join OB. We get ∠BOA whose measure is 60°.
• Constructing a 30° angle:
  1. Construct an angle of 60° as shown earlier. Now, bisect this angle. Each angle is 30°.
• Constructing a 120° angle:
  1. Draw any line PQ and take a point O on it.
  2. Place the pointer of the compasses at O and draw an arc of convenient radius which cuts the line at A.
  3. Without disturbing the radius on the compasses, draw an arc with A as the centre which cuts the first arc at B.
  4. Again without disturbing the radius on the compasses and with B as the centre, draw an arc which cuts the first arc at C.
  5. Join OC, ∠COA is the required angle whose measure is 120°.
• Constructing a 90° angle:
  1. Construct a perpendicular to a line from a point lying on it, as discussed earlier. This is the required 90° angle.
• Constructing a triangle given all three sides:
  Constructing a ΔABC such that AB = 6 cm, BC = 5 cm and CA = 7 cm.
  1. Draw one of the sides say AB = 6 cm.
  2. Using compasses and taking A as the centre, draw an arc of radius 7 cm.
  3. With B as the centre, draw an arc of radius 5 cm, that cuts the first arc at the point C.
  4. Join AC and BC. ΔABC is the required triangle.
• Constructing a triangle when two sides and the included angle is given:
  Constructing a ΔABC such that AB = 5 cm, ∠A = 60° and CA = 8 cm.
  1. Draw AB = 5 cm.
  2. With the help of a compass construct &angPAB = 60°.
  3. With A as centre and radius 5 cm cut AP at point C.
  4. Join BC. ΔABC is the required triangle.
• Constructing a triangle when two angles and included side is given:
  Constructing ΔABC such that AB = 4 cm, ∠A = 30°, ∠B = 60°.
  1. Draw AB = 4 cm.
  2. At A construct ∠QAB = 30°.
  3. At B construct ∠PBA = 60°.
  4. AQ and BP intersect each other at C. ΔABC is the required triangle.
• Constructing the circumcircle of a triangle:

A circle that passes through all the three vertices of a triangle is called the circumcircle of the triangle.

• The point where the perpendicular bisectors of the sides of a triangle meet is called the circumcenter. Here O is the circumcenter.
• OA = OB = OC = radius of circumcircle = Circumradius.
• Constructing the in-circle of a triangle:

A circle drawn inside a triangle such that it touches all the three sides of the triangle is called the in-circle of the triangle.

• The point where the bisectors of the angles of a triangle meet, is called incenter. Here I is the incenter.
• The length of the perpendicular, here IP, is called the inradius.

A polygon is a plane figure with a minimum of three straight lines and three angles.

A triangle is a polygon with three sides. A triangle consists 3 vertices and 3 sides encompassing 3 angles. The sum of the interior angles of a triangle is always 180 degrees.

In this chapter, you will learn about types of triangles and their properties.

Triangles: A triangle is a plane figure bounded by three line segments.

Vertex: Vertex of a triangle is a point where any two of its sides meet.

Angles:

• Sum of interior angles of a triangle is always 180°.
• An interior angle of a triangle can be represented by the letter representing the corresponding vertex.
• When any side of a triangle is extended, the angle formed outside the triangle is called an exterior angle.
• An exterior angle of a triangle is an adjacent and supplementary angle to the corresponding interior angle of a triangle.
• An exterior angle is equal to the sum of opposite interior angles.

Types of triangles according to angles

• If each angle of a triangle is acute i.e. less than 90°, it is called acute angled triangle.
• If each angle of a triangle is equal to  90°, it is called right angled triangle.
  • In a right angled triangle, the side opposite to the right angle is called hypotenuse and it is the largest side of the right angled triangle.             
• If each angle of a triangle is obtuse i.e. greater than 90°, it is called the obtuse angled triangle.

Types of triangles according to sides:

• A triangle with at least two sides equal is called isosceles triangle.
• PQ =PR,so ∠R = ∠P.
• A triangle with all sides equal is called equilateral triangle. Each interior angle of the equilateral triangle = (180°/3) = 90°.
• If the three sides of a triangle are unequal then it is called scalene triangle.

Let us consider an example:

Example:

Find the value of x in the figure given  below:

Solution:

∠ACD = ∠A + ∠B.

115° = 2x + 3x.

4x = 115°.

x = 23°.

It is important to learn construction of triangles. For the construction of a triangle, it is not mandatory to have all its dimensions and angles. A triangle can be constructed given any one of the following set of measurements:

• The length of two sides and measurement of the angle associated.
• Measurements of two angles and length of the side associated.
• For a right-angled triangle, the length of one side and the length of the hypotenuse.
• The length of all the three sides of a triangle.

In this chapter, you will learn construction of triangles with examples.

We shall be constructing a triangle when any of the following conditions are given:

• The length of three sides.
• The length of two sides and angle included between these two sides.
• Any two angles and the included side i.e. the side common to both angles.

Here are some examples:

Example 1:

Construct a triangle ABC such that BC = 4cm, AC = 6cm and AB = 7. cm.

Solution:

Steps:

1) Draw line segment AB = 7.6 cm

2) Taking A as the centre, draw an arc of radius 6 cm.

3)  Taking B as the centre, draw an arc of radius 4 cm which cuts the previous arc at C.

4) Join AC and BC.

ABC is the required triangle. (Refer to the figure on the left side).

Example 2: Construct a triangle ABC such that BC = 5 cm, ∠ABC = 60° and AB = 3cm.

Solution:

Steps:

1) Draw line segment BC  = 5 cm

2) Construct  ∠PBC = 60°

3)  Taking B as the centre, draw an arc of radius 3 cm which cuts BP at A such that BA = 3 cm.

4) Join A and C.

ABC is the required triangle. (Refer to the figure on the right side).

Angles are useful in many situations in day-to-day life.

In this chapter, you will learn about adjacent and vertically opposite angles.

Adjacent angle: Two angles are said to be adjacent angles if,

• They have a common vertex.
• They have one common arm.
• The other arms of the angles are on the opposite sides of the common arm.
• Angle AOB and BOC are adjacent angles.

Vertically opposite angles:

• When two straight lines intersect, the angles on the opposite sides of their point of intersection are called vertically opposite angles.

Congruent angles: Angles having the same angular measurement value are said to be congruent angles.

The word complementary takes its origin from the Latin completum which means “completed” since
the right angle is believed to be a complete angle.  The word supplement comes from Latin supplere, which means to “supply” what is needed.

The two angles which add up to 90 degrees is said to be complementing each other. Similarly, two angles which add up to 180 degrees are said to be supplementary angles.

Complementary angle: When the sum of the measures of two angles is 90°, such angles are called complementary angles and each angle is called a complement of the other.

Supplementary angles: When the sum of the measures of two angles is 180°, such angles are called supplementary angles and each of them is called a supplement of the other.

When two straight lines meet at a point an angle is formed. Angles can be seen every day. When you are sitting in a room, you can see an angle at the meeting point of two walls. When you open a book holding it straight, the two pages are open at a certain angle.

The applications of angles are plenty. They are essential in architecture, to make furniture, clocks and so on.

In this chapter, you will learn about angles.

Angle: Two different rays starting from the same fixed point forms an angle. Symbol ∠ is used to represent an angle.

The measure of the Angle: The amount of turning which one arm must be turned about the vertex to bring it to the position of the other arm is called the measure of an angle.

Interior of the Angle: It is the region that lies within the angle.

Exterior of the Angle: It is the region that lies outside the angle.

The unit of measurement of angles is degrees. ° is used to represent degrees.

A protractor is a semi-circular plastic marked in degrees from 0 to 180° on its semi-circular part is used to measure angles.

Types of angles:

• Complete angle: An angle of measure 360⁰ is called a complete angle.
• Right angle: An angle that measures 90⁰ is called a right angle. A right angle makes a quarter revolutions. (Shown in the left side figure below).
• Straight angle: An angle that measures 180⁰ is called a straight angle. A straight angle makes a half revolution. (Shown in the right side figure below).

• Acute angle: An angle that measures less than 90⁰ is called an acute angle.
• Obtuse angle: An angle that measures more than 90⁰ and less than 180⁰ is called an obtuse angle. (Shown in the left side figure below).
• Reflex angle: An angle that measures more than 180⁰ is called a reflex angle. (Shown in the right side figure below).

Sum of angles around a point is always 360°

You come across many solids every day. The football that you play, the book that you read, the LPG cylinder used for cooking and so on are all solids. Unlike 2D shapes, a solid has length, breadth and depth.

In this chapter, you will learn about solids and recognition of various solid shapes.

Solid: An object that occupies space and has a fixed space is called solid.

Some of the characteristics of a solid are:

• Solids have length, breadth and height. It is a three-dimensional figure.
• Solid objects have only three main views.

• Top view.
• Side view.
• Front view.

Cuboid:

Cuboid is solid or hollow which has 6 rectangular faces.

• It is a three dimensional solid.
• A cuboid has 12 edges.
• A cuboid has 8 vertices. 

Cube:

Cube is a symmetrical three-dimensional shape, either solid or hollow contained by six equal squares.

• Each face of a cube is square.
• Cube has 6 faces.
• Cube has 12 edges.
• Cube has 8 corners.

Cylinder:

Cylinder is a solid or hollow geometrical figure with a curved side and two identical circular flat ends.  

• Cylinder has  2 edges, 3 faces but no vertex.

Sphere:

Sphere is a round solid or hollow figure with every point on its surface equidistant from its centre.

• A sphere is a 3D figure with no vertex, no edges and only1 surface.

Cone:

Cone is a solid or hollow object which tampers from a circular base to a point.

• Cone is a 3D figure with 1 vertex, 1 edge and 2surfaces.

Prism:

Prism is a solid geometrical figure whose two ends are similar, equal and parallel rectilinear figures and whose sides and faces are either parallelograms or rectangles.

• It has 3 faces,9 edges, and 6 vertices.
• Prism is a solid whose side faces are ||gm and whose end basses are two parallel and congruent polygons.

Pyramid:

Pyramid is a solid whose base is a plane rectilinear figure such as triangle and whose side faces are triangles with a common vertex.

• If the base of the pyramid is quadrilateral then it is called a quadrilateral pyramid.
• If the base of the pyramid is triangle then it is called a triangular pyramid.

Euler’s formula:

For a 3-D solid,

• V stands for the number of vertices.
• E stands for the number of edges.
• F stands for the number of faces.

Euler’s formula is V + F – E = 2.

Drawing a cube:

• The two types of sketches for drawing a cube are oblique and isometric.
• An isometric paper has dots or lines, marked on it dividing the paper into small equilateral triangles.

Cube:

Steps:

1. Take a squared paper.
2. Draw the front face.
3. Draw the opposite face of the same size.
4. Join the corresponding corners.
5. Draw the figure with hidden edges dotted.

FUNDAMENTAL GEOMETRICAL CONCEPTS :

The word ‘geometry’ originally come from the Greek word ‘geo’
meaning ‘earth’ and ‘metron’ meaning ‘measurement’.
The word geometry means ‘measurement of earth’ or the science of
properties and relation of figures.

1. Point : A point determine allocation ‘.’
2. Line : A line is a straight path that extends indefinitely in both direction.
3. Intersecting Lines : If two lines have one common point, they are called
   intersecting line

Parallel Lines :

Triangle :

6.Square :

Rectangle :

Polygons :

Circle :

Example : Consider the given figure and answer the following questions :
(i) Is it a curve ?
(ii) Is it a closed figure?

Solution :
(i) It is not a curve . It is a polygon entirely made up of the line
segment .
(ii) Yes, It is closed figure.

Symmetry

When we fold a paper in such a way that the picture is divided into two equal halves then the line which divides the picture into two halves is called a Line of Symmetry.

Here the line divides the star into two halves so it is the line of symmetry. It is also called the Mirror Line because if we place the mirror on that line then one side of the picture will fall exactly on the other side of the picture.

Non-symmetrical Figure

This figure is not symmetrical as if we fold the image from the dotted line then it does not divide it into two equal halves.

Making Symmetric Figures: Ink-blot Devils

To make an ink-blot pattern-

• Take a piece of paper and fold it in half.
• Put some drops of ink on one side of the paper.
• Then press the halves together.
• It will make a symmetric pattern with the fold as the line of symmetry.

Inked-string pattern

To make an inked string pattern-

• Take a piece of paper and fold it in half.
• Dip a string in different colours and arrange it on the one side of the paper.
• Press the two halves together and pull the string.
• It will make a symmetric inked string pattern with the fold as the line of symmetry.

Two Lines of Symmetry

Some figures have two lines of symmetry.

1. A Rectangle

Take a rectangular sheet and fold it horizontally in two equal halves and then again fold it vertically in two equal halves. After opening it, we get two lines of symmetry of the rectangular sheet.

2. More Figures with two Line of Symmetry

If we take a rectangular piece of paper and double fold it to make two lines of symmetry and cut it in some new shape then after opening it we will get a new image that too with the two lines of symmetry.

Construction of figure with two Lines of Symmetry

1. To draw a figure with two lines of symmetry, take one figure.

2. Let L and M be the two lines of symmetry.

3. Draw the figure in such a way that L is the line of symmetry,

4. Now complete the figure by drawing the remaining part so that M will also become the line of symmetry.

Hence this is the final figure with two lines of symmetry.

Multiple Lines of Symmetry

Take a square sheet of paper and fold it in two halves vertically and again horizontally .open it and fold it in two equal halves diagonally then again open it and fold it along another diagonal.

When you will open the paper you will see four imaginary lines and these lines are the lines of symmetry.

Some more images with more than two lines of symmetry

• Equilateral triangle will have three lines of symmetry.
• Square will have four lines of symmetry.
• Regular pentagon will have five lines of symmetry.
• Regular hexagon will have six lines of symmetry.

Some Real-life Examples of Symmetry

In Taj Mahal and the butterfly there is one line of symmetry and there are so many other things also in our daily life which are having one or more line of symmetry.

Reflection and Symmetry

The line of symmetry is also called Mirror Line because the mirror image of an object is symmetrical to the image. When we see an object in the mirror then there is no change in the length and angles of the object except one thing i.e. the image is opposite to the original image.

Some Examples of Reflection Symmetry

1. Paper Decoration

We can use a rectangular sheet to fold and create some intricate patterns by cutting paper.

2. Kaleidoscope

In Kaleidoscope, mirrors are used to create pictures having various lines of symmetry. Two mirrors strips forming a V-shape are used. The angle between the mirrors determines the number of lines of symmetry.

Example

Which alphabet will remain same after reflection symmetry? Check for R, C, N, A and T.

Solution

In the alphabet reflection symmetry, the alphabets look opposite in the mirror i.e. the alphabet written from right to left will appear as written from left to right.

Hence C, N and R will not look the same after reflection.

Hence A and T will look same after reflection symmetry.

Circles

It is a simple closed curve and is not considered as a polygon.

Parts of Circles

1. Radius – It is a straight line connecting the centre of the circle to the boundary of the same. Radii is the plural of ‘radius’.

2. Diameter –It is a straight line from one side of the circle to the other side passing through the centre.

3. Circumference – It refers to the boundary of the circle.

4. Chord – Any line that connects two points on the boundary of the circle is called Chord. Diameter is the longest chord.

5. Arc – It is the portion of the boundary of the circle.

6. Interior of the Circle – Area inside the boundary of the circle is called the Interior of the Circle.

7. The Exterior of the Circle – Area outside the boundary of the circle is called the Exterior of the Circle.

8. Sector– It 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.

9. Segment – It is the region in the interior of the circle enclosed by an arc and a chord.

Semi-circle

A diameter divides the circle into two semi-circles. Hence the semicircle is the half of the circle, which has the diameter as the part of the boundary of the semicircle.

Construction of a circle if the radius is known

Draw a circle of radius 5 cm.

To draw a circle, we need a compass and a ruler to measure the length.

Step 1: Open the compass and measure the length of 5 cm using a ruler.

Step 2: Mark a point O, which we will use as the centre of the circle.

Step 3: Put the pointer on the point O.

Step 4: Turn the compass to make a complete circle. Remember to do it in one instance. 

Prime Factorisation

Prime Factorisation is the process of finding all the prime factors of a number.

There are two methods to find the prime factors of a number-

1. Prime factorisation using a factor tree

We can find the prime factors of 70 in two ways.

The prime factors of 70 are 2, 5 and 7 in both the cases.

2. Repeated Division  Method

Find the prime factorisation of 64 and 80.

The prime factorisation of 64 is 2 × 2 × 2 × 2 × 2 × 2.

The prime factorisation of 80 is 2 × 2 × 2 × 2 × 5.

Highest Common Factor (HCF)

The highest common factor (HCF) of two or more given numbers is the greatest of their common factors.

Its other name is (GCD) Greatest Common Divisor.

Method to find HCF

To find the HCF of given numbers, we have to find the prime factorisation of each number and then find the HCF.

Example

Find the HCF of 60 and 72.

Solution:

First, we have to find the prime factorisation of 60 and 72.

Then encircle the common factors.

HCF of 60 and 72 is 2 × 2 × 3 = 12.

Lowest Common Multiple (LCM)

The lowest common multiple of two or more given number is the smallest of their common multiples.

Methods to find LCM

1. Prime Factorisation Method

To find the LCM we have to find the prime factorisation of all the given numbers and then multiply all the prime factors which have occurred a maximum number of times.

Example

Find the LCM of 60 and 72.

Solution:

First, we have to find the prime factorisation of 60 and 72.

Then encircle the common factors.

To find the LCM, we will count the common factors one time and multiply them with the other remaining factors.

LCM of 60 and 72 is 2 × 2 × 2 × 3 × 3 × 5 = 360

2. Repeated Division Method

If we have to find the LCM of so many numbers then we use this method.

Example

Find the LCM of 105, 216 and 314.

Solution:

Use the repeated division method on all the numbers together and divide until we get 1 in the last row.

LCM of 105,216 and 314 is 2 × 2 × 2 × 3 × 3 × 3 × 5 × 7 × 157 = 1186920

Real life problems related to HCF and LCM

Example: 1

There are two containers having 240 litres and 1024 litres of petrol respectively. Calculate the maximum capacity of a container which can measure the petrol of both the containers when used an exact number of times.

Solution:

As we have to find the capacity of the container which is the exact divisor of the capacities of both the containers, i. e. maximum capacity, so we need to calculate the HCF.

The common factors of 240 and 1024 are 2 × 2 × 2 × 2. Thus, the HCF of 240 and 1024 is 16. Therefore, the maximum capacity of the required container is 16 litres.

Example: 2

What could be the least number which when we divide by 20, 25 and 30 leaves a remainder of 6 in every case?

Solution:

As we have to find the least number so we will calculate the LCM first.

LCM of 20, 25 and 30 is 2 × 2 × 3 × 5 × 5 = 300.

Here 300 is the least number which when divided by 20, 25 and 30 then they will leave remainder 0 in each case. But we have to find the least number which leaves remainder 6 in all cases. Hence, the required number is 6 more than 300.

The required least number = 300 + 6 = 306.

Plane Figures

The closed 2-D shapes are referred to as plane figures.

Here “C” is the boundary of the above figure and the area inside the boundary is the region of this figure. Point D comes in the area of the given figure.

Perimeter

If we go around the figure along its boundary to form a closed figure then the distance covered is the perimeter of that figure. Hence the Perimeter refers to the length of the boundary of a closed figure.

If a figure is made up of line segments only then we can find its perimeter by adding the length of all the sides of the given figure.

Example

Find the Perimeter of the given figure.

Solution

Perimeter = Sum of all the sides

= (12 + 3 + 7 + 6 + 10 + 3 + 15 + 12) m

= 68 m

The Perimeter of a Rectangle

A rectangle is a closed figure with two pairs of equal opposite sides.

Perimeter of a rectangle = Sum of all sides

= length + breadth + length + breadth

Thus, Perimeter of a rectangle = 2 × (length + breadth)

Example: 1

The length and breadth of a rectangular swimming pool are 16 and 12 meters respectively .find the perimeter of the pool.

Solution: 

Perimeter of a rectangle = 2 × (length + breadth)

Perimeter of the pool = 2 × (16 + 12)

= 2 × 28

= 56 meters

Example: 2

Find the cost of fencing a rectangular farm of length 24 meters and breadth 18 meters at 8/- per meter.

Solution:

Perimeter of a rectangle = 2 × (length + breadth)

Perimeter of the farm = 2 × (24 + 18)

= 2 × 42

= 84 meter

Cost of fencing = 84 × 8

= Rs. 672

Thus the cost of fencing the farm is Rs. 672/-.

Regular Closed Figure

Figures with equal length of sides and an equal measure of angles are known as Regular Closed Figures or Regular Polygon.

Perimeter of Regular Polygon = Number of sides × Length of one side

Perimeter of Square

Square is a regular polygon with 4 equal sides.

Perimeter of square = side + side + side + side

Thus, Perimeter of a square = 4 × length of a side

Example

Find the perimeter of a square having side length 25 cm.

Solution

Perimeter of a square = 4 × length of a side

Perimeter of square = 4 × 25

 = 100 cm

Perimeter of an Equilateral Triangle

An equilateral triangle is a regular polygon with three equal sides and angles.

Perimeter of an equilateral triangle = 3 × length of a side

Example

Find the perimeter of a triangle having each side length 13 cm.

Solution

Perimeter of an equilateral triangle = 3 × length of a side

Perimeter of triangle = 3 × 13

= 39 cm

Perimeter of a Regular Pentagon

A regular pentagon is a polygon with 5 equal sides and angles.

Perimeter of a regular pentagon = 5 × length of one side

Example

Find the perimeter of a pentagon having side length 9 cm.

Solution

Perimeter of a regular pentagon = 5 × length of one side

Perimeter of a regular pentagon = 5 × 9

= 45 cm

Perimeter of a Regular Hexagon

A regular hexagon is a polygon with 6 equal sides and angles.

Perimeter of a regular hexagon = 6 × Length of one side

Example

Find the perimeter of a hexagon having side length 15cm.

Solution

Perimeter of a regular hexagon = 6 × Length of one side

Perimeter of a regular hexagon = 6 × 15

= 90 cm

Perimeter of a Regular Octagon

A regular octagon is a polygon with 8 equal sides and angles.

Perimeter of a regular octagon = 8 × length of one side

Example

Find the perimeter of an octagon having side length 7cm.

Solution

Perimeter of a regular octagon = 8 × length of one side

Perimeter of a regular octagon = 8 × 7

= 56 cm

Area

Area refers to the surface enclosed by a closed figure.

To find the area of any irregular closed figure, we can put them on a graph paper with the square of 1 cm × 1 cm .then estimate the area of that figure by counting the area of the squares covered by the figure.

Here one square is taken as 1 sq.unit.

Example

Find the area of the given figure. (1 square = 1 m²)

Solution

The given figure is made up of line segments and is covered with some full squares and some half squares.

Full squares in figure = 32

Half squares in figure = 21

Area covered by full squares = 32 × 1 sq. unit = 32 sq. unit.

Area covered by half squares = 21 × (1/2) sq. unit. = 10.5 sq. unit.

Total area covered by figure = 32 + 10.5 = 42.5 sq. unit.

Area of a Rectangle

Area of a rectangle = (length × breadth)

Example

Find the area of a rectangle whose length and breadth are 20 cm and 12 cm respectively.

Solution

Length of the rectangle = 20 cm

Breadth of the rectangle = 12 cm

Area of the rectangle = length × breadth

= 20 cm × 12 cm

= 240 sq cm.

To find the length of a rectangle if breadth and area are given:

Example

What will be the length of the rectangle if its breadth is 6 m and the area is 48sq.m?

Solution

Length = 48/6

 = 8 m

To find the breadth of the rectangle if length and area are given:

Example

What will be the breadth of the rectangle if its length is 8 m and the area is 81 sq.m?

Solution

Breadth = 81/8

= 9 m

Area of a Square

Area of a square is the region covered by the boundary of a square.

Area of a square = side × side

Example

Calculate the area of a square of side 13 cm.

Solution

Area of a square = side × side

= 13 × 13

= 169 cm².