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

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Let’s do a little review work before discussing similar figures.

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Congruent Figures In order to be congruent, two figures must be the same size and same shape.

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Similar Figures Similar figures must be the same shape, but their sizes may be different.

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Similar Figures This is the symbol that means “similar.” These figures are the same shape but different sizes.

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SIZES Although the size of the two shapes can be different, the sizes of the two shapes must differ by a factor. Factor means multiply, so you must multiply to find scale factor. 3 3 2 1 6 6 2 4

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SIZES In this case, the factor is x 2. 33 2 1 6 6 2 4

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SIZES Or you can think of the factor as 1/2. 33 2 1 6 6 2 4

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Enlargements When you have a photograph enlarged, you make a similar photograph. x 3

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Reductions A photograph can also be shrunk. x 1/4

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Determine the length of the unknown side. 12 9 15 4 3 ?

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These triangles differ by a factor of 3, 12 9 15 4 3 ? 15 x 1/3 = 5 or 1/3.

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Determine the length of the unknown side. 4 2 24 ?

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These dodecagons differ by a factor of 6. 4 2 24 ? 2 x 6 = 12

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Sometimes the factor between 2 figures is not obvious and some calculations are necessary. 18 12 15 12 108 12 x ___ = 18 12 18 = 1.5

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How do you find the scale factor from large to small? 18 12 15 12 108 18 x __ = 12 To find this missing factor, divide 12 by 18. 2/3

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When changing the size of a figure, will the angles of the figure also change? ?? ? 70 40

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Nope! Remember, the sum of all 3 angles in a triangle MUST add to 180 degrees. If the size of the angles were increased, the sum would exceed 180 degrees. 70 40 70 40

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70 40 We can verify this fact by placing the smaller triangle inside the larger triangle. 70 40

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70 40 The 40 degree angles are congruent.

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70 40 The 70 degree angles are congruent.

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70 40 4 The other 70 degree angles are congruent.

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Find the length of the missing side. 30 40 50 6 8 ?

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This looks messy. Let’s translate the two triangles. 30 40 50 6 8 ?

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Now “things” are easier to see. 30 40 50 8 ? 6

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The common factor between these triangles is 5. 30 40 50 8 ? 6

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So the length of the missing side is…?

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That’s right! It’s ten! 30 40 50 8 10 6

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Similarity is used to answer real life questions. Suppose that you wanted to find the height of this tree.

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Unfortunately all that you have is a tape measure, and you are too short to reach the top of the tree.

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

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Then, measure the length of your shadow. 10 feet 2 feet

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

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The tree must be 30 ft tall. Boy, that’s a tall tree! 10 feet 2 feet 6 ft

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Similar figures “work” just like equivalent fractions. Equivalent fractions are called a proportion. 5 30 66 11

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These numerators and denominators differ by a factor of 1/6. 5 30 66 11 x 1/6

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or 6 5 30 66 11 = 6 x

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Similar Figures So, similar figures are two figures that have the same shape, same corresponding angles, and proportionate sides (scale factor).

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Practice Time!

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1) Determine the missing side of the triangle. 3 4 5 12 9 ?

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1) Determine the missing side of the triangle. 3 4 5 12 9 15

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2) Determine the missing side of the triangle. 6 4 6 36 ?

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2) Determine the missing side of the triangle. 6 4 6 36 24

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

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3) Determine the missing sides of the triangle. 39 24 33 13 8 11

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4) Determine the height of the lighthouse. 2.5 8 10 ?

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4) Determine the height of the lighthouse. 2.5 8 10 32

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5) Determine the height of the car. 5 3 12 ?

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5) Determine the height of the car. 5 3 12 7.2

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THE END!

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