(Not exactly the same, but pretty close!) Similar Figures (Not exactly the same, but pretty close!)

Congruent Figures For figures to be congruent, the figures must be the same size and same shape.

What does it mean for a set of figures to be SIMILAR ?

This memorial to Crazy Horse is still in the process of being carved This memorial to Crazy Horse is still in the process of being carved. It was started 55 years ago. When finished, Crazy Horse will be 641 feet long and will stand 563 feet high.                                                                                                                           

Similarity can be used to shrink something into a smaller, more manageable size, or enlarge something into a larger size so that it can be viewed more easily. blueprints for a house enlarging a cell looking at a globe

Similar Figures ~ Similar figures are the same shape but their sizes are different. The symbol for similar is ~

SIZES The sizes of the similar shapes must differ by the same factor. 6 2 4 3 2 1 ~

SIZES In this case, the factor is 2. 4 2 6 6 3 3 1 2

SIZES Or you can think of the factor as 1 2 4 2 6 6 3 3 1 2 4 ∙ 1 2 4 ∙ 1 2 2 6 6 ∙ 1 2 6 3 3 2 ∙ 1 2 1 2

Similarity Transformation-Vocabulary a ________ is a transformation in which a figure and its image are similar. the ratio of the new image to the original figure is called the ____________. a dilation with a scale factor greater than 1 (k>1) is called an ___________. a dilation with a scale factor less than 1 (0<k<1) is called a _________. dilation scale factor, k 𝑘= 𝑖𝑚𝑎𝑔𝑒 𝑝𝑟𝑒𝑖𝑚𝑎𝑔𝑒 enlargement reduction

Enlargements Reductions When you have a photograph enlarged, you make a similar photograph. A photograph can also be shrunk to produce a slide.

The 2 triangles are similar. Determine the missing side length. 15 12 x 4 3 9

Use corresponding sides to set up a proportion: 15 12 4 3 ? 9

Given similar polygons, use corresponding sides to determine the missing side length. x 2 24 4

Find the length of the missing side. This is hard to see each triangle. Let’s translate the them. 50 ? 30 6

Now corresponding sides are easier to see. 50 30 ? 6

Set up a proportion to find the missing side. 50 30 x ? 6

The scale factor for these similar figures is more difficult to guess The scale factor for these similar figures is more difficult to guess. Set up ratios of corresponding sides to help. 15 12 8 10 18 12

18 12 15 10 8 This is called an extended proportion. Remember: 𝑘= 𝑖𝑚𝑎𝑔𝑒 𝑝𝑟𝑒𝑖𝑚𝑎𝑔𝑒 This is called an extended proportion. For the figures to be similar, all 3 ratios must be equal . 10 15 = 8 12 = 12 18 18 12 15 10 8

Similarity can be used to answer real life questions. Suppose that you wanted to find the height of this tree. Unfortunately all that you have is a tape measure, and you are too short to reach the top of the tree.

You can measure the length of the tree’s shadow. 10 feet

Then, measure the length of Sam’s shadow. 10 feet 2 feet

If you know how tall Sam is, then you can determine how tall the tree is. or 6 ft 10 feet 2 feet

In summary: Similar figures have the same shape congruent corresponding angles proportional corresponding sides

3) Determine the missing sides of the triangle. 39 y 33 x 8 24

How do the perimeters of similar figures relate to each other? Find the perimeter of each triangle. 3 4 5 9 12 15 P = 3 + 4 + 5 p = 12 P = 9 + 12 + 15 p = 36 Find the scale factor from the green to the blue triangle. 𝑘= 9 3 = 3 1 𝑟= 36 12 = 3 1 Now find the ratio of their perimeters.

How do the areas of similar figures relate to each other? Find the area of each triangle. 3 4 5 9 12 15 𝐴= 1 2 ∙3∙4 𝐴= 1 2 ∙9∙12 𝐴=6 𝐴=54 The scale factor from the green to the blue triangle is 𝑘= 3 1 𝑟= 54 6 Find the ratio of their areas. = 9 1

In summary: If figures are similar figures then The ratio of their perimeters is equal to the ratios of their corresponding sides. (k) The ratio of their areas is equal to the ratio of their corresponding sides squared. (k2)

Given the scale factor from the larger to the smaller triangle is 1 3 : Find the perimeter of the smaller triangle. P(large ∆)=12+15+9 = 36 P(small ∆) = 36 ∙ 1 3 = 12 12 9 15 b) Find the area of the smaller triangle. A(large ∆) = 1 2 ∙9∙12 = 54 A(small ∆) = 54∙ 1 3 2 = 54∙ 1 9 = 6