Unit 8 Similarity Ratios, Proportions, Similar Polygons Geometry w/Ms. Hernandez St. Pius X High School

Unit 8 Similarity This unit is about similar polygons. Two polygons are similar if their corresponding angles are congruent and the lengths of corresponding sides are proportional. Some vocabulary: ratio, proportion, means, extremes, geometric mean, similar polygons, scale factor

8.1 Ratio If a and b are two quantities that are measured in the SAME units, then the ratio of a to b is a : b. Ratios can be written 3 different ways. a to b a : b

8.1 Ratio You must be able to simplify ratios Example In this example we used the fact that there are 100 cm in 1 m. So then 4m = 4 x 100cm = 400cm. Next, we reduced 12/400. 12 and 400 both have the factor 4 in common. So we can factor out a 12 from the numerator and denominator.

8.1 Ratio You must be able to USE ratios Example The perimeter of a rectangle ABCD is 60 centimeters. The ratio of AB:BC is 3:2. Find the length and width of the rectangle. At first glance it may look like we don’t have enough information to solve this problem. Let’s take apart the given information and see how we can solve it.

8.1 Using Ratios perimeter of a rectangle ABCD is 60 centimeters The perimeter of a rectangle ABCD is 60 centimeters. The ratio of AB:BC is 3:2. Find the length and width of the rectangle. perimeter of a rectangle ABCD is 60 centimeters Since we are working with rectangle ABCD let’s draw it first.

8.1 Using Ratios perimeter of a rectangle ABCD is 60 centimeters. The perimeter of a rectangle ABCD is 60 centimeters. The ratio of AB:BC is 3:2. Find the length and width of the rectangle. perimeter of a rectangle ABCD is 60 centimeters. The perimeter of rectangle is found from the formula P = 2l + 2w where l = length and w = width. So we then could write 60cm = 2l + 2w

8.1 Using Ratios ratio of AB:BC is 3:2 The perimeter of a rectangle ABCD is 60 centimeters. The ratio of AB:BC is 3:2. Find the length and width of the rectangle. ratio of AB:BC is 3:2 Remember we can write ratios in different ways, therefore we can write the ratios as fractions

8.1 Using Ratios Find the length and width of the rectangle The perimeter of a rectangle ABCD is 60 centimeters. The ratio of AB:BC is 3:2. Find the length and width of the rectangle. Find the length and width of the rectangle AB represents length in this rectangle and BC represents width in this rectangle So we can write AB = 3x and BC = 2x

8.1 Using Ratios Putting it all together 60cm = 2(3x) + 2(2x) The perimeter of a rectangle ABCD is 60 centimeters. The ratio of AB:BC is 3:2. Find the length and width of the rectangle. Putting it all together 60cm = 2(3x) + 2(2x) 60cm = 6x + 4x 60cm = 10x 6cm = x L = AB = 3x = 3(6) = 18 W = BC = 2x = 2(x) =12

8.1 Using Ratios This same technique can be applied in area problems and measures of interior angles of a triangle. In section 8.1 see example #3 in your textbook and problems #29-32 in your textbook.

8.1 Extended Ratios BC:AC:AB is 3:4:5 Solve for x. So then we write the following ratios And we can use any of these ratios to solve for x.

8.2 Problem Solving in Geometry with Proportions We can take advantage of the following properties of proportions to being solving problems in Geometry.

8.2 Examples In the diagram AB = AC. Find the length of . BD CE

8.2 More examples Use your textbook to complete more examples. Sometimes x is not in the picture and you have to figure out where it goes. Try these problems from section 8.2 #23-28

8.2 Geometric Mean The geometric mean of two numbers a and b is x and is represented by the following proportion. Find the geometric mean of 3 and 27.

8.2 Geometric Mean Another way to respsent the geometric mean of two numbers a and b is. Find the geometric mean of 3 and 27.