Basic Geometrical Ideas PDF

# 9 Basic Geometrical Ideas ## Learning Objectives - To identify a point, line, line segment and ray - To differentiate between open and closed figures - To identify curves, angles, triangles, quadrilaterals and circles - To identify parts of a triangle, quadrilateral and circle ## Let's Get St...

# 9 Basic Geometrical Ideas ## Learning Objectives - To identify a point, line, line segment and ray - To differentiate between open and closed figures - To identify curves, angles, triangles, quadrilaterals and circles - To identify parts of a triangle, quadrilateral and circle ## Let's Get Started Does your school have a play area? Can you name the sport for which the given play area is used? - Is the length of the various lines and curves in the field fixed? - Identify some geometrical shapes inside it. - In which subject do we study about the measurement of various play areas? The branch of Mathematics in which we study about various shapes and their measurements is called Geometry. The word Geometry is derived from the Greek word ‘Geometron’. ‘Geo’ means Earth and ‘metron’ means ‘measurement’. Geometry was invented to measure land and other shapes. # Introduction to Geometry ABCD represents a rectangular sheet of paper. - Does this paper have thickness? Is it flat or curved? - Are the edges straight or curved? What can you say about AB, BC, CD and AD? AB and AD are meeting at A. What is A? Make a small dot with pencil on this sheet and name it X. - Make another small dot Y, a little away from X. - Do the dots X and Y have length, breadth or thickness? Are they just marking a location on the sheet? - Place your pencil on X and without lifting it, move it to reach Y. What kind of path did you follow to go from X to Y? Was it straight, curved or zig-zag? Mark a point M just below X, and a point N just below Y. Join M and N with the help of a ruler. - Which distance is shorter: XY or MN? Draw a straight path PQ, which can be extended in either direction if needed. - Mark a point O in your notebook. Starting from O, draw a straight path which can be continued infinitely and name it OR. ## Definitions * **Point:** A point marks a location and has no length, breadth and thickness. The dots A, B, C, D, X, Y, M, N and O marked in the above figure are points. Point is named with one capital letter of the English alphabet. The sharp end of a thin needle and the ends of a divider are examples of points. * **Plane:** The surface of a solid may be curved like that of a cricket ball or it may be flat like the top of a table. A plane is a flat surface that extends infinitely in all directions It has length and breadth, but no thickness. A sheet of paper and a green board in a classroom are flat surfaces with limited length and breadth and are a part of the plane which extends infinitely in all directions. Two planes meet in a straight line. For example, two adjacent walls of a room meet to form the height of a room. * **Line:** There are infinite points on a plane. A line is a continuous collection of points that forms a straight path. Infinite lines can be drawn passing through a given point. A line extends on both sides indefinitely. It has no endpoints, and thus, has no definite length. A line is denoted by two capital letters or by a single small letter. * **Line Segment:** A line segment is the shortest distance between two fixed endpoints. There is only one line segment which can join two given points. A line segment is denoted by two capital letters. * **Ray:** A ray is a part of a line with a fixed starting point and continues infinitely in the other direction. It does not have a definite length. | **Line** | **Line Segment** | **Ray** | |-----------------|-----------------------------|----------------------------------------------| | A line is a straight path that continues infinitely in both directions. It does not have a definite length. A line is denoted by AB. | A line segment is a part of a line which has a fixed starting point and endpoint. It has a definite length. A line segment is denoted by AB. | A ray is a part of a line with a fixed starting point and continues infinitely in the other direction. It does not have a definite length. A ray is denoted by AB. | ## Collinear and Non-Collinear Points - **Collinear points:** Two points in a plane can always be joined by a straight line. Three or more points in a plane, which can be joined by a straight line, are called collinear points. P, Q and R are collinear points. - **Non-collinear points:** Three or more points in a plane, which cannot be joined by a straight line, are called non-collinear points. A, B and C are non-collinear points and form a triangle. ## Open and Simple Closed Figures - **Open figure:** An open figure is a figure that starts at a point but does not end at the same point. - **Simple closed figure:** A simple closed figure is a figure that starts and ends at the same point. It does not intersect itself and has a closed boundary. A simple closed figure has points which lie in its interior, points on the boundary and points in its exterior. In the adjoining figure, E and F are points in the interior of the closed figure, H is in the exterior and G is on the boundary of the figure. - **Curvilinear figures:** are figures whose boundaries are made up of only curved lines. For example, circles and ovals are curvilinear figures. - **Linear figures:** are figures whose boundaries are made up of only straight line segments. For example, squares and rectangles are linear figures. | **Open figure** | **Simple closed figure** | |-------------------------|-----------------------------| | Starting and endpoints are different. Such figures are made by lines or curves. Open figures are incomplete shapes. | Starting and endpoints are the same. Such figures are made by lines or curves that do not cross each other. Closed figures are complete shapes, like polygons or circles. | # Exercise 9A 1. Identify the following in the figure given here. - Line segments **AB, BC, DA, CD** - 3 collinear points **A, D, C** - 3 non-collinear points **A, B, C** 2. Fill in the blanks: - The shortest distance between two fixed **points** is the line segment. - A closed figure starts and ends at the same point. - The boundaries of linear figures are made of only straight line segments. 3. State whether the following statements are true or false: - A line has no breadth or thickness. **True** - A circle is an open figure. **False** - A triangle is a linear figure. **True** # Angle Mark a point O on a piece of paper. Draw rays OA and OB, with O as the initial point (as shown in the figure). The figure made by two intersecting rays is called an angle. The point where the two rays meet is the vertex of the angle. The two rays are called the arms or sides or legs of the angle. An angle is denoted by the symbol ∠. An angle can be named by one capital letter (∠O), one small letter (∠x) or by three capital letters of the English alphabet (∠AOB or ∠BOA). In the given figure, O is the vertex, OA and OB are the arms of the angle and ∠O or ∠AOB or ∠BOA or ∠x is the angle. When we name an angle with three letters, the vertex is written in the middle. We can name an angle with one letter when only one angle is formed at a vertex, else we need to name the angle with three capital letters. Since two angles are formed at the vertex O in the adjoining figure, we name the angles as ∠AOB and ∠AOC. A few examples where the angles are found in real life are shown below. The unit used for measuring an angle is degree. A degree is represented by the symbol ° A degree can be divided into 60 equal parts called minutes, 1° = 60’. A minute is divided into 60 equal parts called seconds, 1’ = 60”. # Interior and Exterior of an Angle The space which lie within the arms of an angle form the interior of the angle. The space which do not lie within the arms of an angle form the exterior of the angle. Some points lie on the arms of an angle. The point P is in the interior of ∠AOB, the point Q is in the exterior of ∠AOB, and the points A, O, B and R lie on ∠AOB. # Triangle Three points in a plane, lying on a straight line, are said to be collinear. Three non-collinear points in a plane form a triangle. Tri means 3. Triangle means three angles. A triangle is a closed figure made of three line segments. It has three angles and three vertices. Vertex of a triangle is a point where two of its sides meet. Plural of vertex is vertices. ## Parts of a Triangle A triangle has six parts. Its parts are the three sides and the three angles. - Sum of the angles of a triangle is 180°. - Interior of a triangle consists of all points which lie inside the boundary of the triangle. - Exterior of a triangle consists of all points which lie outside the boundary of the triangle. - Boundary of a triangle consists of all points which lie on the sides of the triangle. In the given triangle, - Vertices: A, B and C - Sides: AB, AC and BC - Angles: ∠A, ∠B and ∠C or ∠BAC, ∠ABC and ∠ACB or ∠CAB, ∠CBA and ∠BCA - Side opposite to vertex A = BC - Side opposite to vertex B = AC - Side opposite to vertex C = AB - ∠A + ∠B + ∠C = 180° - Points in the interior: P and Q - Points in the exterior: X and Y - Points on the triangle: A, B, C and Z # Quadrilateral A quadrilateral is a closed figure with four sides. A quadrilateral has four vertices, four sides, four angles and two diagonals. When we name a quadrilateral, we name it either clockwise or anticlockwise, with letters in sequence. - Sum of the angles of a quadrilateral is 360°. - ∠E + ∠F + ∠G + ∠H = 360° - Interior of a quadrilateral consists of all points which lie inside the boundary of the quadrilateral. - Exterior of a quadrilateral consists of all points which lie outside the boundary of the quadrilateral. - Boundary of a quadrilateral consists of all points which lie on the sides of the quadrilateral. - Opposite sides are the sides which do not have a common vertex. - Opposite angles are the angles which do not have a common arm. - Adjacent sides are the sides which have a common vertex. In the given quadrilateral, - Vertices: A, B, C and D - Sides: AB, BC, CD and DA - Angles: ∠DAB, ∠CBA, ∠BCD and ∠ADC - Diagonals: AC and BD - Opposite sides: AD and BC, AB and DC - Opposite angles: ∠DAB and ∠BCD, ∠ADC and ∠ABC - Adjacent sides: AB and BC, AB and AD, AD and DC, BC and DC - ∠DAB + ∠ABC + ∠BCD + ∠ADC = 360° - Points in the interior: P and Q - Points in the exterior: X and Y - Points on the quadrilateral: A, B, C, D and Z # Exercise 9B 1. Identify the following in the figure given here: - Triangles **AOB, BOC, ABC, ADC** - Quadrilateral **ABCD** - A diagonal of the quadrilateral **AC** - Opposite sides of the quadrilateral **AB, CD; BC, AD** - Adjacent sides of the quadrilateral **AB-BC, BC-CD, CD-DA, DA-AB** - Opposite angles of the quadrilateral **∠A and ∠C, ∠B and ∠D** - A point in the interior of the quadrilateral **O** - A point in the exterior of the quadrilateral **E** 2. Fill in the blanks: - An angle is formed when two **rays** or **lines** meet. - The parts of a triangle are its **sides** and **vertices**. - A **diagonal** of a quadrilateral joins non-consecutive vertices. 3. State whether the following statements are true or false: - Only two angles can be formed at any vertex. **False** - A triangle has two vertices. **False** 4. Identify the following in the given figure: - Angle opposite to PQ in triangle PQR. **∠QRP** - Angle opposite to SR in triangle PSR. **∠SPR** - Side opposite to PQ in quadrilateral PQRS. **RS** - Side opposite to PS in quadrilateral PQRS. **QR** - Angle opposite to side PR in triangle PQR. **∠PQR** - Angle opposite to side PR in triangle PSR. **∠PSR** - Side opposite to angle P in triangle PSR. **QS** - Side opposite to angle P in triangle PQR. **QR** # Circle A circle is a simple closed curve all of whose points in a plane are at a constant distance from a given fixed point. - **Centre:** The fixed point is called the centre of the circle. - **Circumference:** The boundary or perimeter of a circle is called its circumference. - **Radius:** of a circle is a line segment joining the centre of the circle to any point on the circumference. All radii of a circle are equal. Plural of radius is radii. There are infinite radii of a circle. - **Chord:** of a circle is a line segment joining any two points on the circumference. - **Secant:** is a line passing through two points on the circumference of a circle. - **Diameter:** of a circle is a line segment joining two points on the circumference and passing through the centre of the circle. Diameter of a circle is the longest chord of the circle. Diameter of a circle is twice its radius. There are infinite diameters of a circle. In the given figure, - Radii: OA, OB and OC - Points in the interior: O - Diameter: AB; Secant: XY - Chord: PQ and AB - Points in the exterior: X and Y - Points on the circle: P, Q, A, B and C # Segments - **Segments:** A chord of a circle divides it into two parts, called segments. - **Semicircle:** The diameter of the circle divides it into two equal parts. Each of the equal parts is called a semicircle. A chord, other than diameter, divides a circle into a minor and a major segment. - **Minor segment:** If a chord of a circle divides it into a part which is less than half of the circle, the part is called a minor segment. - **Major segment:** If a chord of a circle divides it into a part which is more than half of the circle, the part is called a major segment. The centre of the circle is a part of the major segment of the circle. ## Arc of a Circle - **Arc of a circle:** is a part of the circumference of a circle. - **Minor arc:** An arc of a circle, which is less than half the circumference of the circle, is called a minor arc. - **Major arc:** An arc of a circle, which is more than half the circumference of the circle, is called a major arc. - **Sector:** The region between an arc and the bounding radii, which join the endpoints of the arc to the centre of a circle, is called a sector. - **Minor sector:** The region between a minor arc and the bounding radii of a circle is called a minor sector. - **Major sector:** The region between a major arc and the bounding radii of a circle is called a major sector. - **Concentric circles:** Two or more circles with the same centre but different radii are called concentric circles. # Exercise 9C 1. Identify the following in the figure given here: - Radii **OA, OB, OC** - Diameter **AC** - Chords **AB, BC, CD, AD, AC** 2. Colour the following parts in the figure: - A minor segment **yellow** - A major segment **pink** - The circumference **green** - A minor arc **blue** 3. Shade the following parts in the figure: - A minor sector with broken lines - A major sector with dots 4. Fill in the blanks: - The boundary of a circle is called its **circumference**. - Diameter of a circle is **twice** its radius. - Concentric circles have the same **centre**. - The region between a minor arc and the bounding radii of a circle is called a **sector**. - **Arc** is a part of the circumference of a circle. - A chord is a line segment but a secant is a **line**. - Diameter of a circle divides it into two **semicircles**. - The longest chord in a circle is the **diameter**. 5. State whether the following statements are true or false: - A quadrilateral has four diagonals. **False** - The minor arc of a circle is greater than the major arc. **False** - A secant is a line segment joining any two points on the circumference. **True** - Every chord of a circle is its diameter. **False** # Maths Lab Activity ## Aim To draw a house with a fence by using various available shapes ## Materials Required - Small circular objects, like coins of different sizes. - Set squares. - Ruler. - One A4 size white sheet - Pencils and colour pencils - String/wool - Scissors and glue stick ## Procedure 1. Make a group of 2 students. 2. Using the above objects, draw and colour a house with fencing. The drawing should have the following: - Triangular shapes - Quadrilaterals - Circular and semicircular shapes - An open figure # Concept Map ## Key Concepts - **Line:** A straight path that continues infinitely in both directions. - **Line Segment:** A part of a line with a fixed starting and endpoint. - **Ray:** A part of a line which has fixed starting point and continues infinitely in the other direction. - **Collinear points:** Three or more points in a plane which can be joined by a straight line. - **Open figure:** Starting and endpoints are different. - **Simple closed figure:** Starting and endpoints are same. - **Curvilinear figures:** Boundaries are curved lines. - **Linear figures:** Boundaries are straight line segments. - **Angle:** Figure made by two intersecting rays. - **Triangle:** A closed figure made of three line segments. - **Quadrilateral:** A closed figure with four sides. - **Circle:** A simple closed curve, all of whose points in a plane are at a constant distance from a given point. # Chapter Revision ## Choose the Correct Option 1. The maximum number of line segments in the given figure is **d. six** 2. The one of lines which can be drawn passing through a given point is **d. infinite** 3. The sum of angles of a triangle is **b. 180°** 4. The sum of angles of a quadrilateral is **d. 360°** 5. The number of diameters of a circle is **d. infinite** ## Fill in the Blanks 1. A line segment which extends infinitely in one direction is called a **ray**. 2. A **point** has no length, breadth or thickness. 3. A **line** has no endpoints. 4. The standard unit of measuring an angle is **degree**. 5. A triangle has **3** parts. ## State Whether the Following Statements are True or False 1. Only one line can be drawn passing through two given points. **False** 2. Three non-collinear points form a triangle. **True** 3. Flat surfaces are plane surfaces. **True** 4. The length of a line can be measured. **False** 5. Opposite angles of all quadrilaterals are equal. **False** ## Match the Following 1. Line **c. Collinear points** 2. Adjacent sides **e. Common vertex** 3. Circle **b. Diameter** 4. Quadrilateral **a. Diagonal** 5. Triangle **d. Three non-collinear points** # Skill Up! ## A. Integration with Physical Education Find out about the different geometric shapes used in the field of sports. Make a report on your findings. You can include the following points in your report - **Which shapes are used to make a basketball court and a badminton court?** - **Will the angle at which the basketball is thrown towards the basket determine whether a basket is scored?** - **Will the angle at which the shuttlecock is hit towards the side of the opponent determine whether a smash occurs?** ## B. Is Team Spirit Necessary Is team spirit necessary while playing a doubles game of badminton or a game of basketball? Why? ## C. Maximum Number of Triangles and Line Segments Find the maximum number of triangles and line segments in the figure given below. How many circles can you draw using the line segments as the diameter of a circle? # Sustainable Development Goals Question B can lead to the exchange of views on **SDG 3 Good Health and Well-Being**. Some amount of physical activity is a must to keep a fit body and mind. What are the health benefits in particular? How does playing a sport improve emotional and mental well-being?