Copyright © 2011 Pearson Education, Inc. Publishing as Prentice Hall. Chapter 7 Graphs and Triangle Applications

Copyright © 2011 Pearson Education, Inc. Publishing as Prentice Hall. 7.1 Reading Pictographs, Bar Graphs, Histograms, and Line Graphs

Martin-Gay, Prealgebra & Introductory Algebra, 3ed 33 Copyright © 2011 Pearson Education, Inc. Publishing as Prentice Hall. Pictographs A pictograph is a graph in which pictures or symbols are used. This type of graph contains a key that explains the meaning of the symbol used. An advantage of using a pictograph to display information is that comparisons can easily be made. A disadvantage of using a pictograph is that it is often hard to tell what fractional part of a symbol is shown.

Martin-Gay, Prealgebra & Introductory Algebra, 3ed 44 Copyright © 2011 Pearson Education, Inc. Publishing as Prentice Hall. The pictograph shows the top eight most-spoken (primary) languages. Source: Pictographs

Martin-Gay, Prealgebra & Introductory Algebra, 3ed 55 Copyright © 2011 Pearson Education, Inc. Publishing as Prentice Hall. Bar Graphs Bar graphs can appear with vertical bars or horizontal bars. An advantage to using bar graphs is that a scale is usually included for greater accuracy.

Martin-Gay, Prealgebra & Introductory Algebra, 3ed 66 Copyright © 2011 Pearson Education, Inc. Publishing as Prentice Hall. The bar graph shows the number of different endangered species. Source: U.S. Fish and Wildlife Service Bar Graphs

Martin-Gay, Prealgebra & Introductory Algebra, 3ed 77 Copyright © 2011 Pearson Education, Inc. Publishing as Prentice Hall. The bar graph shows the population of the world’s largest cities. Source: United Nations, Dept. for Economic and Social Information and Policy Analysis Bar Graphs

Martin-Gay, Prealgebra & Introductory Algebra, 3ed 88 Copyright © 2011 Pearson Education, Inc. Publishing as Prentice Hall. Histograms A histogram is a special bar graph. The width of each bar represents a range of numbers called a class interval. The height of each bar corresponds to how many times a number in the class interval occurred and is called the class frequency. The bars in a histogram lie side by side with no space between them.

Martin-Gay, Prealgebra & Introductory Algebra, 3ed 99 Copyright © 2011 Pearson Education, Inc. Publishing as Prentice Hall. Histograms The test scores of 36 students are summarized in the table Student Scores Frequency (# of students)

Martin-Gay, Prealgebra & Introductory Algebra, 3ed 10 Copyright © 2011 Pearson Education, Inc. Publishing as Prentice Hall. Line Graphs Another common way to display information graphically is by using a line graph. An advantage of a line graph is that it can be used to visualize relationships between two quantities. A line graph can also be very useful in showing change over time.

Martin-Gay, Prealgebra & Introductory Algebra, 3ed 11 Copyright © 2011 Pearson Education, Inc. Publishing as Prentice Hall. Line Graphs Source: National Climatic Data Center Temperatures

Copyright © 2011 Pearson Education, Inc. Publishing as Prentice Hall. 7.2 Reading and Drawing Circle Graphs

Martin-Gay, Prealgebra & Introductory Algebra, 3ed 13 Copyright © 2011 Pearson Education, Inc. Publishing as Prentice Hall. Circle Graphs A circle graph is often used to show percents in different categories, with the whole circle representing 100%.

Martin-Gay, Prealgebra & Introductory Algebra, 3ed 14 Copyright © 2011 Pearson Education, Inc. Publishing as Prentice Hall. The following graph shows the percent of visitors to the United States in a recent year by various regions. Circle Graphs Source: Office of Travel and Tourism Industries

Martin-Gay, Prealgebra & Introductory Algebra, 3ed 15 Copyright © 2011 Pearson Education, Inc. Publishing as Prentice Hall. Drawing Circle Graphs To draw a circle graph, we use the fact that a whole circle contains 360° (degrees). 360°

Martin-Gay, Prealgebra & Introductory Algebra, 3ed 16 Copyright © 2011 Pearson Education, Inc. Publishing as Prentice Hall. The following table shows the percent of U.S. armed forces personnel that are in each branch of the service. Branch of ServicePercent Army35% Navy25% Marine Corps12% Air Force26% Coast Guard2% Drawing Circle Graphs

Martin-Gay, Prealgebra & Introductory Algebra, 3ed 17 Copyright © 2011 Pearson Education, Inc. Publishing as Prentice Hall. To draw a circle graph showing this data, we find the number of degrees in each sector representing each branch of service. Sector Degrees in Each Sector Army35% 360° = 126° Navy25% 360° = 90° Marine Corps 12% 360° ≈ 43° Air Force 26% 360° ≈ 94° Coast Guard 2% 360° ≈ 7° Drawing Circle Graphs

Martin-Gay, Prealgebra & Introductory Algebra, 3ed 18 Copyright © 2011 Pearson Education, Inc. Publishing as Prentice Hall. We draw a circle and mark its center. Then we draw a line from the center of the circle to the circle itself. We use a protractor to construct the sectors. We place the hole in the protractor over the center of the circle. Then we adjust the protractor so that 0° on the protractor is aligned with the line that we drew. Drawing Circle Graphs

Martin-Gay, Prealgebra & Introductory Algebra, 3ed 19 Copyright © 2011 Pearson Education, Inc. Publishing as Prentice Hall. To construct the “Army” sector, we find 126° on the protractor and mark our circle. Then we remove the protractor and use this mark to draw a second line from the center to the circle itself. Drawing Circle Graphs

Martin-Gay, Prealgebra & Introductory Algebra, 3ed 20 Copyright © 2011 Pearson Education, Inc. Publishing as Prentice Hall. To construct the “Navy” sector next, we follow the same procedure as before except that we line 0° up with the second line we drew and mark the protractor this time at 90°. Drawing Circle Graphs

Martin-Gay, Prealgebra & Introductory Algebra, 3ed 21 Copyright © 2011 Pearson Education, Inc. Publishing as Prentice Hall. We continue in this manner until the circle graph is complete. Drawing Circle Graphs

Copyright © 2011 Pearson Education, Inc. Publishing as Prentice Hall. 7.3 Square Roots and the Pythagorean Theorem

Martin-Gay, Prealgebra & Introductory Algebra, 3ed 23 Copyright © 2011 Pearson Education, Inc. Publishing as Prentice Hall. The square of a number is the number times itself. The square of 6 is 36 because 6 2 = 36. The square of –6 is also 36 because The Square of a Number (–6) 2 = (–6) (–6) = 36.

Martin-Gay, Prealgebra & Introductory Algebra, 3ed 24 Copyright © 2011 Pearson Education, Inc. Publishing as Prentice Hall. A square root of 36 is 6 because A square root of 36 is 6 because 6 2 = 36. A square root of 36 is also 6 because ( A square root of 36 is also –6 because (–6) 2 = 36. Square Root of a Number We use the symbol, called a radical sign, to indicate the positive square root of a nonnegative number. because 4 2 = 16 and 4 is positive. because 5 2 = 25 and 5 is positive. The reverse process of squaring is finding a square root.

Martin-Gay, Prealgebra & Introductory Algebra, 3ed 25 Copyright © 2011 Pearson Education, Inc. Publishing as Prentice Hall. Square Root of a Number The square root,, of a positive number a is the positive number b whose square is a. In symbols,

Martin-Gay, Prealgebra & Introductory Algebra, 3ed 26 Copyright © 2011 Pearson Education, Inc. Publishing as Prentice Hall. Remember that the radical sign is used to indicate the positive square root of a nonnegative number. Helpful Hint

Martin-Gay, Prealgebra & Introductory Algebra, 3ed 27 Copyright © 2011 Pearson Education, Inc. Publishing as Prentice Hall. Numbers like are called perfect squares because their square root is a whole number or a fraction. Perfect Squares

Martin-Gay, Prealgebra & Introductory Algebra, 3ed 28 Copyright © 2011 Pearson Education, Inc. Publishing as Prentice Hall. A square root such as cannot be written as a whole number or a fraction since 6 is not a perfect square. It can be approximated by estimating, by using a table, or by using a calculator. Approximating Square Roots

Martin-Gay, Prealgebra & Introductory Algebra, 3ed 29 Copyright © 2011 Pearson Education, Inc. Publishing as Prentice Hall. One important application of square roots has to do with right triangles. A right triangle is a triangle in which one of the angles is a right angle or measures 90º (degrees). The of a right triangle is the side opposite the right angle. The hypotenuse of a right triangle is the side opposite the right angle. hypotenuse leg The of a right triangle are the other two sides. The legs of a right triangle are the other two sides. Right Triangles

Martin-Gay, Prealgebra & Introductory Algebra, 3ed 30 Copyright © 2011 Pearson Education, Inc. Publishing as Prentice Hall. Pythagorean Theorem If a and b are the lengths of the legs of a right triangle and c is the length of the hypotenuse, then In other words, c a b (leg) 2 + (other leg) 2 = (hypotenuse) 2.

Copyright © 2011 Pearson Education, Inc. Publishing as Prentice Hall. 7.4 Congruent and Similar Triangles

Martin-Gay, Prealgebra & Introductory Algebra, 3ed 32 Copyright © 2011 Pearson Education, Inc. Publishing as Prentice Hall. Two triangles are congruent when they have the same shape and the same size. Corresponding angles are equal, and corresponding sides are equal. Congruent Triangles a = 6c = 11 b = 9 d = 6 e = 11 f = 9 equal angles

Martin-Gay, Prealgebra & Introductory Algebra, 3ed 33 Copyright © 2011 Pearson Education, Inc. Publishing as Prentice Hall. Angle-Side-Angle (ASA) If the measures of two angles of a triangle equal the measures of two angles of another triangle, and the lengths of the sides between each pair of angles are equal, the triangles are congruent. Angle-Side-Angle (ASA) These two triangles are congruent by Angle-Side-Angle.

Martin-Gay, Prealgebra & Introductory Algebra, 3ed 34 Copyright © 2011 Pearson Education, Inc. Publishing as Prentice Hall. Side-Side-Side (SSS) If the lengths of the three sides of a triangle equal the lengths of the corresponding sides of another triangle, the triangles are congruent. Side-Side-Side (SSS) These two triangles are congruent by Side-Side-Side.

Martin-Gay, Prealgebra & Introductory Algebra, 3ed 35 Copyright © 2011 Pearson Education, Inc. Publishing as Prentice Hall. Side-Angle-Side (SAS) If the lengths of two sides of a triangle equal the lengths of corresponding sides of another triangle, and the measures of the angles between each pair of sides are equal, the triangles are congruent. Side-Angle-Side (SAS) These two triangles are congruent by Side-Angle-Side.

Martin-Gay, Prealgebra & Introductory Algebra, 3ed 36 Copyright © 2011 Pearson Education, Inc. Publishing as Prentice Hall. Similar triangles are found in art, engineering, architecture, biology, and chemistry. Two triangles are similar when they have the same shape but not necessarily the same size. Similar Triangles

Martin-Gay, Prealgebra & Introductory Algebra, 3ed 37 Copyright © 2011 Pearson Education, Inc. Publishing as Prentice Hall. In similar triangles, the measures of corresponding angles are equal and corresponding sides are in proportion. a = 3 c = 8 b = 5 d = 6e = 10 f = 16 Side a corresponds to side d, side b corresponds to side e, and side c corresponds to side f. Similar Triangles

Martin-Gay, Prealgebra & Introductory Algebra, 3ed 38 Copyright © 2011 Pearson Education, Inc. Publishing as Prentice Hall. In similar triangles, the measures of corresponding angles are equal and corresponding sides are in proportion. a = 3 c = 8 b = 5 d = 6e = 10 f = 16 Side a corresponds to side d, side b corresponds to side e, and side c corresponds to side f. Similar Triangles

Martin-Gay, Prealgebra & Introductory Algebra, 3ed 39 Copyright © 2011 Pearson Education, Inc. Publishing as Prentice Hall. Similar Triangles Example: Find the length of the side labeled n of the following pair of similar triangles. Solution: Since the triangles are similar, corresponding sides are in proportion. Thus, the ratio of 8 to 14 is the same as the ratio of 9 to n. The length of the missing side is units. 8 9n 14

Martin-Gay, Prealgebra & Introductory Algebra, 3ed 40 Copyright © 2011 Pearson Education, Inc. Publishing as Prentice Hall. Finding Unknown Lengths of Sides in Similar Triangles EXAMPLE SOLUTION Find the length of the side labeled n of the following pair of similar triangles. 8 9 n 14 Since the triangles are similar, corresponding sides are in proportion. Thus, the ratio of 8 to 14 is the same as the ratio of 9 to n.

Copyright © 2011 Pearson Education, Inc. Publishing as Prentice Hall. 7.5 Counting and Introduction to Probability

Martin-Gay, Prealgebra & Introductory Algebra, 3ed 42 Copyright © 2011 Pearson Education, Inc. Publishing as Prentice Hall. Likelihood or Probability In our daily conversations, we often talk about the likelihood or the probability of a given result occurring for a chance happening. We call the chance happening an experiment. The possible results of an experiment are called outcomes.

Martin-Gay, Prealgebra & Introductory Algebra, 3ed 43 Copyright © 2011 Pearson Education, Inc. Publishing as Prentice Hall. Using a Tree Diagram Flipping a coin is an experiment and the possible outcomes are heads (H) or tails (T) and are equally likely to happen. One way to picture the outcomes of an experiment is to draw a tree diagram. Each outcome is shown on a separate branch. For example, the outcomes of flipping a coin are H T

Martin-Gay, Prealgebra & Introductory Algebra, 3ed 44 Copyright © 2011 Pearson Education, Inc. Publishing as Prentice Hall. A Tree Diagram for Tossing a Coin Twice There are 4 possible outcomes when tossing a coin twice. H T H T H T First TossSecond TossOutcomes H,H H,T T,H T,T

Martin-Gay, Prealgebra & Introductory Algebra, 3ed 45 Copyright © 2011 Pearson Education, Inc. Publishing as Prentice Hall. The Probability of an Event probability of an event  number of ways that the event can occur number of possible outcomes To find the probability of an event, divide the number of ways that the event can occur by the number of possible outcomes.

Martin-Gay, Prealgebra & Introductory Algebra, 3ed 46 Copyright © 2011 Pearson Education, Inc. Publishing as Prentice Hall. Note from the definition of probability that the probability of an event is always between 0 and 1, inclusive (i.e., including 0 and 1). A probability of 0 means an event won’t occur, and a probability of 1 means that an event is certain to occur. Probability of an Event