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## Similar Figures

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**Similar Figures**(Not exactly the same, but pretty close!)**Let’s do a little review work before discussing similar**figures.**Congruent Figures**• In order to be congruent, two figures must be the same size and same shape.**Similar Figures**• Similar figures must be the same shape, but their sizes may be different.**Similar Figures**This is the symbol that means “similar.” These figures are the same shape but different sizes.**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. 4 2 6 6 3 3 2 1**SIZES**• In this case, the factor is x 2. 4 2 6 6 3 3 2 1**SIZES**• Or you can think of the factor as 1/2 . 4 2 6 6 3 3 2 1**Enlargements**• When you have a photograph enlarged, you make a similar photograph. x 3**Reductions**A photograph can also be shrunk. x 1/4**Determine the length of the unknown side.**15 12 ? 4 3 9**These triangles differ by a factor of 3,**or 1/3. 15 x 1/3 = 5 15 12 ? 4 3 9**These dodecagons differ by a factor of 6.**? 2 x 6 = 12 2 24 4**Sometimes the factor between 2 figures is not obvious and**some calculations are necessary. 15 12 8 10 18 12 12 x ___ = 18 12 18 = 1.5**How do you find the scale factor from large to small?**To find this missing factor, divide 12 by 18. 15 12 8 10 18 12 18 x __ = 12 2/3**When changing the size of a figure, will the angles of the**figure also change? ? 40 70 ? ? 70**Nope! Remember, the sum of all 3 angles in a triangle MUST**add to 180 degrees.If the size of theangles were increased,the sum would exceed180degrees. 40 40 70 70 70 70**We can verify this fact by placing the smaller triangle**inside the larger triangle. 40 40 70 70 70 70**The 40 degree angles are congruent.**40 70 70 70 70**The 70 degree angles are congruent.**40 40 70 70 70 70 70**The other 70 degree angles are congruent.**4 40 70 70 70 70 70**Find the length of the missing side.**50 ? 30 6 8 40**This looks messy. Let’s translate the two triangles.**50 ? 30 6 8 40**Now “things” are easier to see.**50 30 ? 6 40 8**The common factor between these triangles is 5.**50 30 ? 6 40 8**That’s right! It’s ten!**50 30 10 6 40 8**Similarity is 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.**Then, measure the length of your shadow.**10 feet 2 feet**If you know how tall you are, then you can determine how**tall the tree is. 6 ft 10 feet 2 feet**The tree must be 30 ft tall. Boy, that’s a tall tree!**6 ft 10 feet 2 feet**Similar figures “work” just like equivalent fractions.**Equivalent fractions are called a proportion. 30 5 11 66**These numerators and denominators differ by a factor of 1/6.**30 x 1/6 5 x 1/6 11 66**or 6**30 = 6 x 5 11 66 = 6 x**Similar Figures**• So, similar figures are two figures that have the same shape, same corresponding angles, and proportionate sides (scale factor).**1) Determine the missing side of the triangle.**? 9 5 3 4 12**1) Determine the missing side of the triangle.**15 9 5 3 4 12**2) Determine the missing side of the triangle.**36 36 6 6 4 ?**2) Determine the missing side of the triangle.**36 36 6 6 4 24**3) Determine the missing sides of the triangle.**39 ? 33 ? 8 24**3) Determine the missing sides of the triangle.**39 13 33 11 8 24**4) Determine the height of the lighthouse.**? 8 2.5 10**4) Determine the height of the lighthouse.**32 8 2.5 10**5) Determine the height of the car.**? 3 5 12**5) Determine the height of the car.**7.2 3 5 12