UNIT 6: SIMILARITY. When you resize a figure it gets bigger or smaller, but it still looks similar. 14 cm. 5 cm 45º

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1 UNIT 6: SIMILARITY. Similar Figures: Two figures are similar if they are identical in shape but not in size. Examples: Two circles are always similar Two squares are always similar If one figure can become another using resizing (also called dilation, contraction, compression, enlargement or expansion), then the figures are similar. When you resize a figure it gets bigger or smaller, but it still looks similar. Look at these similar polygons: 2 cm 4 cm 7 cm 225º 4 cm 14 cm 225º 8 cm 45º 5 cm 45º The second figure is an enlargement of the first one. The lengths have doubled, but the angles have stayed the same. For any pair of similar figures: corresponding angles are equal, corresponding sides are in the same ratio. This ratio is called similarity ratio. If the similarity ratio of two similar figures is r, then: the ratio of their perimeters is also r, the ratio of their areas is r 2 the ratio of their volumes is r cm 1

2 Your Turn 1. The rectangles below are similar. Find the unknown side of the second rectangle, the similarity ratio, and compare their areas. 4 cm 10 cm 6 cm 2. The following prisms are similar. Find the unknown sides of the second prism, the similarity ratio, and compare their volumes. 2 cm 10 cm 4 cm 14 cm 2

3 3. Which trapezium below is similar to the trapezium DEFG? P Q D E 4 6 G 12 F S 14 R J K W X 3 5 Z 9 Y M 10 L 4. Kristin wants to resize a 4-inch wide by 5-inch long photograph for the school newspaper. It is to fit in a space that is 2 inches long. What is the width of the resized photograph? 4 in. 2 in. 5 in. 5. On a map, two cities are 2,5 cm apart. The scale of the map is 1: What is the actual distance between the towns? 6. A model house is 12 cm wide. If it was built with a scale of 3 cm : 4 m, how wide would the real house be? 3

4 7. Polygon ABCD is similar to polygon EFGH. Each side of polygon ABDC is 3,25 times longer than the corresponding side of polygon EFGH. Find the perimeter of polygon ABCD. A B E 2 in. F 3 in. D H 3 in. 5 in. G C 8. The perimeter for two similar octagons is 3:2. The larger octagon has an area of 75 cm 2. What is the area of the smaller octagon? 9. Lydia plans to use a photocopy machine to increase the size of a small chart that she has made as part of her science project. The original chart is 4 inches by 5 inches. If she uses a scale factor of 5:11, will the chart fit in a piece of paper 8,5 inches by 11 inches? 10. The scale of a model is 1:250. Find: a) The real height and diameter of a cylindrical tower that are 6 and 4 cm respectively in the model. b) The real area of a garden that is 40 cm 2 in the model. c) The real volume of a swimming pool that contains 20 cm 3 of water in the model. 4

5 Thales Theorem:If two non parallel straight lines intersect with parallel straight lines, the ratios of any two segments on the first line are equal to the ratios of the corresponding segments on the second line. C' B' A' A B C That is: AB BC = A' B ' B ' C ', you can also say that AB A' B ' = BC B ' C ', or even you can say: AB A' B ' = BC B ' C ' = AC A' C ' Your Turn 1. Calculate x, y, z: z 3 cm 5 cm y 9 cm x 4 cm 5 cm 5

6 2. Calculate x e y: 3 cm 3 cm 4 cm y 5 cm Similar triangles: In similar triangles: corresponding pairs of angles are equal: A= A' ; B= B ' ; C= C ' a corresponding pairs of sides are in the same ratio a' = b b' = c c' A c B A' c' B' b' b a' a C' C Criteria for similarity of triangles:two triangles are similar if they have all their corresponding pair of angles equal and corresponding sides are in the same ratio. But we don't have to know all three sides and all three angles... two or three out of the six is enough. There are three ways to know if two triangles are similar: Criterion 1: If two triangles have two of their angles equal, the triangles are similar. Criterion 2: If two triangles have two pair of sides in the same ratio, then the triangles are similar. Criterion 3: If two triangles have two pair of sides in the same ratio and the included angles are also equal, then the triangles are similar. 6

7 Example: In the following figure the segments DE and AB are parallel. Prove that the triangles ABC and CDE are similar and find the length of DE and EC. B E 4,8 cm 8,4 cm C A 6 cm D 12 cm Similarity in right-angled triangles: Criterion 1: Two right-angled triangles are similar if of the first is equal to one of the acute angles of the second. Criterion 2: Two right-angled triangles are similar if their legs are proportional. Your Turn 1. Old Faithful in Yellowstone National Park shoots water 60 feet into the air that casts a shadow of 42 feet. What is the height of a nearby tree casts a shadow 63 feet long? Assume that the triangles are similar. (Old Faithful is a cone geyser located in Wyoming, in Yellowstone National Park in the United States). 7

8 2. A flagpole casts a 6 m shadow. At the same time, Humberto, who is 1,80 m tall, casts a 1,5 m shadow. What is the height of the flagpole? B 3. a) Prove that the triangles ABC and AED are similar. b) Find the perimeter of the trapezium EBCD. A 17 cm E 6 cm D 10 cm C 4. If you extend the two non parallel sides of this trapezium until they intersect, you get two triangles in Thales' position. Find the perimeter of the larger triangle. 15 cm 13 cm 20 cm The altitude of the hypotenuse of a right-angled triangle forms two triangles that are similar to each other and to the original triangle. ABC and AMB are similar because they share the angle B. ABC and AMC are similar because they share the angle C. 8

9 Theorem of the leg: Since these triangles are similar, we can establish proportions relating the corresponding sides. The triangles ABC and MAC are similar, so a b = b m b2 =a m The triangles ABC and MBA are similar, so a c = c n c2 =a n 9

10 Theorem of the altitude: The triangles MAC and MBA are similar, so m n = h n h2 =m n To sum up: Theorem of the leg: b 2 =a m c 2 =a n Theorem of the altitude: h 2 =m n 10

11 Your Turn 1. Find the value of x and y in each of the following right-angled triangles: 2. A leg of a right triangle is 12 m. Its projection on the hypotenuse is 7,2 cm. Find the perimeter and the area of this triangle. 11

12 3. The height of a cone inscribed in a sphere is 12 cm. If the radius of the sphere is 15 cm, find the lateral area of the cone. 4. Look at this picture. The man's eye are 1,7 m above the ground, the distance from the man to the edge of the well is 0,8 m and the width of the well is 1,2 m. What is the depth of the well? 5. Liam, who is 1,65 m tall, wants to determine the height of a building. He stands 1,5 m from the fence and takes the measurements shown in the picture. What is the height of the building? 12

13 Keywords: Similarity = Semejanza similar figures = figuras semejantes shape = forma size = tamaño similar ratio /scale factor = razón de semejanza scale = escala Thales theorem = teorema de Tales criterion / criteria = criterio / criterios Pythagoras theorem = teorema de Pitágoras right-angled triangle / right triangle = triángulo rectángulo hypotenuse = hipotenusa leg = cateto segment = segmento Theorem of the leg = teorema del cateto Theorem of the altitude = teorema de la altura 13