unit 3 three geometric ratio final-1.pptx

geometric ratio final

1. Chapter 3: Geometric Ratio Compiled by: Ms Naveen

2. What Are Ratios ? A ratio is the relationship between two or more quantities among a group of items. The purpose of a ratio is find the relationship between two or more items in the collection. Let's look at an example. Here is a collection of coins. There are four pennies and two dimes, for a total of six coins. The ratio of pennies to dimes can be expressed as a ratio, and a ratio can be written in three different, but equivalent ways. One of the ways to express ratios is as a fraction. Do you see it?

3. When expressed as fractions, ratios can then be rewritten as fractions in simplest form. Notice that the ratio of pennies to dimes is 4 to 2, which, as a fraction, can be written as 2 to 1 . For some groupings, you can write a ratio that includes three terms .

4. Ratios usually include similar types of objects. Here is a collection of different types of balls.

5. What Are Rates ? Rates are a special type of ratio. Rates involve different types of quantities. Here is an example of a rate involving cost and weight. 

6. What Are Equivalent Ratios ? Ratios express a part-to-part relationship between two items in a group of items. Equivalent ratios result when the ratios expressed as fractions are equal. Let's look at an example. This is a collection of colored circles. First, look at the ratio of green : yellow. 2 : 4 Now look at the ratio of orange : red. 1 : 2 But when written as fractions 1/2 and 2/4 are equivalent fractions. This also means they are equivalent ratios. 

7. What Are Proportions ? When two ratios are equivalent, they form a proportion. In the previous section you saw that the ratio green : yellow was equivalent to orange : red. Because they are equivalent ratios, they form a proportion. When two quantities are proportional, then corresponding parts are proportional. Let's look at an example. When measurement ratios like this are equivalent, then the rectangles are proportional. Another way of saying this is that the rectangles are similar. When two geometric figures are similar, that means they have the same shape, but not necessarily the same size.

8. Definition of Proportion Proportion  refers to the equality of two ratios. Two equivalent ratios are always in proportion. Proportions are denoted by the symbol (: :) and they help us to solve for unknown quantities. In other words, proportion is an equation or statement that is used to depict that the two ratios or  fractions  are equivalent. Four non-zero quantities, a, b, c, d are said to be in proportion if a : b = c : d. Now , let us consider the two ratios 3 : 5 and 15 : 25. Here , 3 : 5 can be expressed as 3:5 = 3/5 = 0.6 and 15:25 can be expressed as 15:25 = 15/25 = 3/5 = 0.6. Since both the ratios are equal, we can say that these two are proportional .

9. T ypes of proportions. There are two types of proportions. Direct Proportion Inverse Proportion Direct Proportion Direct proportion describes the direct relationship between two quantities. If one quantity increases, the other quantity also increases and vice-versa. Thus, a direct proportion is written as y ∝ x. For example, if the speed of a car is increased, then it covers more distance in a fixed period of time.

10. Inverse Proportion Inverse proportion describes the relationship between two quantities in which if one quantity increases, the other quantity decreases and vice-versa. Thus, an inverse proportion is written as y ∝ 1/x. For example, as the speed of a vehicle is increased, it will cover a fixed distance in less time.

11. Ratio and Proportion Formula The formula for ratio is expressed as a : b ⇒ a/b, where, a = the first term or antecedent. b = the second term or consequent. For example, ratio 2 : 7 is also represented as 2/7, where 2 is the antecedent and 7 is the consequent.

12. Now , in order to express a proportion for the two ratios, a : b and c : d, we write it as a:b::c:d⟶ ab=cd The two terms b and c are called mean terms. The two terms a and d are known as extreme terms . In a: b = c : d, the quantities a and b should be of the same kind with the same units , whereas, c and d may be separately of the same kind and of the same units. For example, 5 kg : 15 kg = Rs . 75 : Rs . 225 In a proportion , the product of the means = the product of the extremes. Therefore , in the  proportion formula  a: b : : c : d, we get b × c = a × d. For example, in 5 : 15 :: 75 : 225, we will get 15 × 75 = 5 × 225

13. Difference Between Ratio and Proportion The difference between ratio and proportion can be seen in the following table. Ratio Proportion It is used to compare the size of two quantities with the same unit. It is used to express the relation of two ratios. The symbols used to express a ratio - a colon (:), slash (/) The symbol used to express a proportion - double colon (::) It is referred to as an  expression . It is referred to as an  equation .

14. Examples on Ratio and Proportion Example 1:  Find out if the ratios 6:8 and 24:32 are in proportion. Solution: The given ratios are 6:8 and 24:32. 6:8= 3/4 = 0.75 and 24:32 = 3/4= 0.75. Here, both the ratios are equal. Therefore, 6:8 and 24:32 are in proportion.

15. Example 2:   There are 30 students in a class. The number of students who like Math and the ones who like Science is expressed in the ratio 2:3. Find the number of students who like Math and the ones who like Science. Solution : Total number of students = 30. Let the number of students who like Math = 2x the number of students who like Science = 3x. We can say that 2x + 3x = 30  ⇒  5x = 30  ⇒  x = 6. Substituting the value of x = 6, we get the numbers of students who like Math = 2x = 2 × 6 = 12 and the number of students who like Science = 3x = 3 × 6 = 18. Therefore, 12 students like Math and 18 students like Science.

16. Example 3:  Given ratio are- a:b = 2:3 b:c = 5:2 c:d = 1:4 Find a: b: c . Solution : Multiplying the first ratio by 5, second by 3 and third by 6, we have a:b = 10: 15 b:c = 15 : 6 c:d = 6 : 24 In the ratio’s above, all the mean terms are equal, thus a:b:c:d = 10:15:6:24

17. Example 4: The earnings of Rohan is 12000 rupees every month and Anish is 191520 per year. If the monthly expenses of every person are around 9960 rupees. Find the ratio of the savings. Solution :  Savings of Rohan per month = Rs (12000-9960) = Rs . 2040 Yearly income of Anish = Rs . 191520 Hence, the monthly income of Anish = Rs . 191520/12 = Rs . 15960. So, the savings of Anish per month = Rs (15960 – 9960) = Rs . 6000 Thus, the ratio of savings of Rohan and Anish is Rs . 2040: Rs.6000 = 17: 50.

18. Example 6: The dimensions of the rectangular field are given. The length and breadth of the rectangular field are 50 meters and 15 meters. What is the ratio of the length and breadth of the field? Solution:  Length of the rectangular field = 50 m Breadth of the rectangular field = 15 m Hence, the ratio of length to breadth = 50: 15 ⇒ 50: 15 = 10: 3. Thus, the ratio of length and breadth of the rectangular field is 10:3.

19. Home Task Simplify the ratio 16:4. Simplify the ratio 75:45:30:60. Are the ratios 4:5 and 8:10 said to be in Proportion? Are the two ratios 8:10 and 7:10 in proportion? Check whether the following statements are true or false. a] 12 : 18 = 28 : 56 b] 25 people : 130 people = 15kg : 78kg There exists 45 people in an office. Out of which female employees are 25 and the remaining are male employees. Find the ratio of a] The count of females to males. b] The count of males to females.

20. Linear Functions

21. Linear Functions In Mathematics, a linear function is defined as a function that has either one or two variables without exponents. It is a function that graphs to the straight line. In case, if the function contains more variables, then the variables should be constant, or it might be the known variables for the function to remain it in the same linear function condition.

22. What is a Linear Function? A  linear function  is a function which forms a straight line in a graph. It is generally a polynomial function whose degree is utmost 1 or 0.  Although the linear functions are also represented in terms of calculus as well as  linear algebra . Linear Function Graph has a straight line whose expression or formula is given by; y=f(x)= px+q It has one independent and one dependent variable. The independent variable is x and the dependent one is y. P is the constant term or the y-intercept and is also the value of the dependent variable. When x = 0, q is the coefficient of the independent variable known as slope which gives the rate of change of the dependent variable.

23. What is a Nonlinear Function? A function which is not linear is called nonlinear function. In other words, a function which does not form a straight line in a graph. The examples of such functions are  exponential function , parabolic function, inverse functions, quadratic function, etc. The expression for all these functions is different.

24. Linear Function Graph Graphing a linear equation involves three simple steps: Firstly, we need to find the two points which satisfy the equation, y = px+q . Now plot these points in the graph or X-Y plane. Join the two points in the plane with the help of a straight line.

25. Linear Function Formula The expression for the linear function is the formula to graph a straight line. The expression for the linear equation is; y = mx + c where m is the slope, c is the intercept and ( x,y ) are the coordinates. This formula is also called slope formula. While in terms of function, we can express the above expression as; f(x) = a x + b, where x is the independent variable.

26. Linear Function Characteristics Let’s move on to see how we can use function notation to graph 2 points on the grid. Relation : It is a group of ordered pairs. Variable : A symbol that shows a quantity in a math expression. Linear function : If each term is either a constant or It is the product of a constant and also (the first power of) a single variable, then it is called as an algebraic equation. Function : A function is a relation between a set of inputs and a set of permissible outputs. It has a property that each input is related to exactly one output. Steepness : The rate at which a function deviates from a reference Direction : Increasing, decreasing, horizontal or vertical.

27. Coordinate Graphs Coordinate geometry deals with graphing (or plotting) and analyzing points, lines, and areas on the coordinate plane (coordinate graph). Each point on a number line is assigned a number. In the same way, each point in a plane is assigned a pair of numbers. These numbers represent the placement of the point relative to two intersecting lines. In  coordinate graphs two perpendicular number lines are used and are called  coordinate axes.   One axis is horizontal and is called the  x ‐axis.  The other is vertical and is called the  y ‐axis.  The point of intersection of the two number lines is called the  origin  and is represented by the coordinates (0, 0).

28. Each point on a plane is located by a unique  ordered pair  of numbers called the  coordinates.  Some coordinates are noted in Figure 2.

29. The coordinate graph is divided into four quarters called  quadrants.  These quadrants are labeled in Figure 3.  Coordinate graph with quadrants labeled. Notice the following: In quadrant I,  x  is always positive and  y  is always positive.  In quadrant II,  x  is always negative and  y  is always positive.  In quadrant III,  x  and  y  are both always negative.  In quadrant IV,  x  is always positive and  y  is always negative.

30. Example 1 Graph the equation  x  +  y  = 6. 

31. Notice that these solutions, when plotted, form a straight line. Equations whose solution sets form a straight line are called  linear equations.  Complete the graph of  x  +  y  = 6 by drawing the line that passes through these points.  Equations that have a variable raised to a power, show division by a variable, involve variables with square roots, or have variables multiplied together will not form a straight line when their solutions are graphed. These are called  nonlinear equations.

32. Symmetry

33. Symmetry And Line Of Symmetry: An Introduction In geometry, symmetry is defined as a balanced and proportionate similarity that is found in two halves of an object. It means one-half is the mirror image of the other half. The imaginary line or axis along which you can fold a figure to obtain the symmetrical halves is called the  line of symmetry . If an object is symmetrical, it means that it is equal on both sides. Suppose, if we fold a paper such that half of the paper coincides with the other half of the paper, then the paper has symmetry.

34. Symmetry can be defined for both regular and irregular shapes. For example, a square is a regular (all sides are equal) and a rectangle is an irregular shape (since only opposite sides are equal). The symmetries for both shapes are different. Check  different figures with symmetry  here.

35. Symmetry in Mathematics In Mathematics, a meaning of symmetry defines that one shape is exactly like the other shape when it is moved, rotated, or flipped. Consider an example, when you are told to cut out a ‘heart’ from a piece of paper, don’t you simply fold the paper, draw one-half of the heart at the fold and cut it out to find that the other half exactly matches the first half? The heart carved out is an example of symmetry.

36. Symmetry Math definition states that “symmetry is a mirror image”. When an image looks identical to the original image after the shape is being turned or flipped, then it is called symmetry. It exists in patterns. You may have often heard of the term ‘symmetry’ in day to day life. It is a balanced and proportionate similarity found in two halves of an object, that is, one-half is the mirror image of the other half. And a shape that is not symmetrical is referred to as asymmetrical. Symmetric objects are found all around us, in nature, architecture, and art.

37. Symmetrical Figures Symmetrical shapes  or figures are the objects where we can place a line such that the images on both sides of the line mirror each other.  The below set of figures form symmetrical shapes when we place a plane or draw the lines.

38. For example, figure (b) has the  symmetrical figures  when we draw two lines of symmetry as shown below .

39. Line of Symmetry The imaginary line or axis along which you fold a figure to obtain the symmetrical halves is called the line of symmetry. It basically divides an object into two mirror-image halves. The line of symmetry can be vertical, horizontal or diagonal. There may be one or more lines of symmetry. 1 Line Symmetry Figure is symmetrical only about one axis. It may be horizontal or vertical. The word ATOYOTA has one axis of symmetry along the axis passing through Y.

40. 2 Lines  Symmetry Figure is symmetrical with only about two lines. The lines may be vertical and horizontal lines as viewed in the letters H and X. Thus, we can see here  two lines symmetry .

41. 3 Lines Symmetry An example of three lines of symmetry is an equilateral triangle. Here, the mirror line passes from the vertex to the opposite side dividing the triangle into two equal right triangles.

42. 4 Lines Symmetry  Four lines of symmetry can be seen in a square, that has all the sides equal. 

43. Infinite lines Some figures have not one or two, but infinite lines passing through the centre , and the figure is still symmetrical. Example: a circle.

44. Types of Symmetry Symmetry may be viewed when you flip, slide or turn an object. There are four types of symmetry that can be observed in various situations, they are: Translation Symmetry Rotational Symmetry Reflection Symmetry Glide Symmetry

45. Translation Symmetry If the object is translated or moved from one position to another, the same orientation in the forward and backward motion is called translational symmetry. In other words, it is defined as the sliding of an object about an axis. This can be observed clearly from the figure given below, where the shape is moved forward and backward in the same orientation by keeping the fixed axis .

46. Rotational Symmetry When an object is rotated in a particular direction, around a point, then it is known as rotational symmetry or radial symmetry. Rotational symmetry existed when a shape turned, and the shape is identical to the origin. The angle of rotational symmetry is the smallest angle at which the figure can be rotated to coincide with itself. The order of symmetry is how the object coincides with itself when it is in  rotation . In geometry, many shapes consist of rotational symmetry. For example, the figures such as circle, square, rectangle have rotational symmetry. Rotational symmetry can also be found in nature, for instance, in the petals of a flower. Below figure shows the rotational symmetry of a square along with the degree of rotation.

47. Reflexive Symmetry   Reflection symmetry is a type of symmetry in which one half of the object reflects the other half of the object. It is also called mirror symmetry or line of symmetry. A classic example of reflection symmetry can be observed in nature, as represented in the below figure.

48. Glide Symmetry The combination of both translation and reflection transformations is defined as the glide reflection. A glide reflection is commutative in nature. If we change the combination’s order, it will not alter the output of the glide reflection.

49. Symmetrical Shapes The symmetry of shapes can be identified whether it is a line of symmetry, reflection or rotational based on the appearance of the shape.  The shapes can be regular or irregular. Based on their regularity, the shapes can have symmetry in different ways.  Also, it is possible that some shapes does not have symmetry. For example, a tree may or may not have symmetry.

50. Surface Areas and Volume

51. Surface Areas and Volume Surface area and volume  are calculated for any three-dimensional geometrical shape. The surface area of any given object is the area or region occupied by the surface of the object. Whereas  volume  is the amount of space available in an object .

52. In geometry, there are different shapes and sizes such as sphere, cube, cuboid, cone, cylinder, etc. Each shape has its surface area as well as volume. But in the case of  two-dimensional figures  like square, circle, rectangle, triangle, etc., we can measure only the area covered by these figures and there is no volume available. Now, let us see the formulas of surface areas and volumes for different 3d-shapes.

53. What is Surface Area ? The space occupied by a two-dimensional flat surface is called the area. It is measured in square units. The area occupied by a three-dimensional object by its outer surface is called the surface area. It is also measured in square units. Generally, Area can be of two types: (i) Total Surface Area (ii) Curved Surface Area/Lateral Surface Area

54. Types of S urface Area Total Surface Area Total surface area refers to the area including the base(s) and the curved part. It is the total area covered by the surface of the object. If the shape has a curved surface and base, then the total area will be the sum of the two areas. Curved Surface Area/Lateral Surface Area Curved surface area refers to the area of only the curved part of the shape excluding its base(s). It is also referred to as lateral surface area for shapes such as a cylinder.

55. What is Volume ? The amount of space, measured in cubic units, that an object or substance occupies is called volume. Two-dimensional doesn’t have volume but has area only. For example, the  Volume of the Circle  cannot be found, though the Volume of the sphere can be. It is so because a sphere is a three-dimensional shape.

56. Surface Area and Volume Formulas Name Perimeter Total Surface Area Curved Surface Area Volume Figure Square 4b b 2 —- —- Rectangle 2(w+h) w.h —- —-

57. Name Perimeter Total Surface Area Curved Surface Area Volume Figure Parallelogram 2( a+b ) b.h —- —- Ellipse 2π√( a 2  + b 2 )/2       π a.b —- —- Trapezoid a+b+c+d 1/2(a+b).h —- —- Circle 2 π r π  r 2 —- —-

58. Name Perimeter Total Surface Area Curved Surface Area Volume Figure Triangle a+b+c 1/2 * b * h —- —- Cuboid 4(l+b+h) 2(lb+bh+hl) 2h(l+b) l * b * h Cube 6a  6a 2 4a 2 a 3 Cylinder —- 2 π  r(r+h) 2π rh π  r 2  h

59. Name Perimeter Total Surface Area Curved Surface Area Volume Figure Cone —- π  r(r+l) π  r l 1/3π r 2  h Sphere —- 4 π r 2 4π r 2 4/3π r 3 Hemisphere —- 3 π r 2 2 π r 2 2/3π r 3  

60. Solved Examples Example 1: What is the surface area of a cuboid with length, width and height equal to 4.4 cm, 2.3 cm and 5 cm, respectively? Solution : Given , the dimensions of cuboid are: length , l = 4.4 cm width, w = 2.3 cm height , h = 5 cm Surface area of cuboid = 2( wl+hl+hw ) = 2·(2.3 x 4.4 + 5 x 4.4 + 5 x 2.3) = 87.24 square cm.

61. Example 2: What is the volume of a cylinder whose base radii are 2.1 cm and height is 30 cm? Solution : Given , Radius of bases, r = 2.1 cm Height of cylinder = 30 cm Volume of cylinder = πr 2 h =  π·(2.1) 2 ·30 ≈ 416.

62. Practice Questions Find the volume of a cube whose side length is 5 cm. Find the CSA of the hemisphere, if the radius is 7 cm. If the radius of the sphere is 4 cm, find its surface area.

63. Accuracy and Precision - The Art of Measurement Measurement is essential for us to understand the external world, and through millions of years of life, we have developed a sense of measurement. Measurements require tools that provide scientists with a quantity. The problem here is that the result of every measurement by any measuring instrument contains some uncertainty. This uncertainty is referred to as an error. Accuracy and precision are two important factors to consider while taking measurements. Both these terms reflect how close a measurement is to a known or accepted value. In this article, let us learn in detail about precision and accuracy.

64. Accuracy The ability of an instrument to measure the accurate value is known as accuracy. In other words, it is the  the closeness of the measured value to a standard or true value . Accuracy is obtained by taking small readings. The small reading reduces the error of the calculation. The accuracy of the system is classified into three types as follows : Point Accuracy The accuracy of the instrument only at a particular point on its scale is known as point accuracy. It is important to note that this accuracy does not give any information about the general accuracy of the instrument.

65. Accuracy as Percentage of Scale Range The uniform scale range determines the accuracy of a measurement. This can be better understood with the help of the following example: Consider a thermometer having the scale range up to 500 ºC . The thermometer has an accuracy of ±0.5 percent of scale range i.e. 0.005 x 500 = ± 2.5 ºC. Therefore, the reading will have a maximum error of ± 2.5 ºC . Accuracy as Percentage of True Value Such type of accuracy of the instruments is determined by identifying the measured value regarding their true value. The accuracy of the instruments is neglected up to ±0.5 percent from the true value.

66. Precision The closeness of two or more measurements to each other is known as the precision of a substance. If you weigh a given substance five times and get 3.2 kg each time, then your measurement is very precise but not necessarily accurate. Precision is independent of accuracy. The below examples will tell you about how you can be precise but not accurate and vice versa. Precision is sometimes separated into : Repeatability The variation arising when the conditions are kept identical and repeated measurements are taken during a short time period. Reproducibility The variation arises using the same measurement process among different instruments and operators, and over longer time periods.

67. Difference between Precision and Accuracy Accuracy  Precision Accuracy refers to the level of agreement between the actual measurement and the absolute measurement. Precision implies the level of variation that lies in the values of several measurements of the same factor. Represents how closely the results agree with the standard value. Represents how closely results agree with one another. Single-factor or measurement are needed. Multiple measurements or factors are needed to comment about precision. It is possible for a measurement to be accurate on occasion as a fluke. For a measurement to be consistently accurate, it should also be precise. Results can be precise without being accurate. Alternatively, the results can be precise and accurate.

68. Examples Q1 ) The volume of a liquid is 26 mL. A student measures the volume and finds it to be 26.2 mL, 26.1 mL, 25.9 mL, and 26.3 mL in the first, second, third, and fourth trial, respectively. Which of the following statements is true for his measurements? a. They are neither precise nor accurate. b. They have poor accuracy. c. They have good precision. d. They have poor precision. Answer:  They have good precision. Q2 ) The volume of a liquid is 20.5 mL. Which of the following sets of measurement represents the value with good accuracy? 18.6 mL, 17.8 mL, 19.6 mL, 17.2 mL 19.2 mL, 19.3 mL, 18.8 mL, 18.6 mL 18.9 mL, 19.0 mL, 19.2 mL, 18.8 mL 20.2 mL, 20.5 mL, 20.3 mL, 20.1 mL Answer:  The set 20.2 mL, 20.5 mL, 20.3 mL, 20.1 mL represents the value with good accuracy.