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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 shapes • Are Enlargements of each other • Corresponding angles are equal • Sides are related by the same scale factor 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 shapes4must differ by a factor. 2 3 1 3 6 6 2 SIZES • In this case, the factor is x 2. 4 2 3 3 1 6 6 2 SIZES • Or you can think of the factor as 2. 4 2 3 3 1 6 6 2 Enlargements • When you have a photograph enlarged, you make a similar photograph. X3 Reductions • A photograph can also be shrunk to produce a slide. 4 Determine the length of the unknown side. 15 12 ? 4 3 9 These triangles differ by a factor of 3. 15 15 12 3= 5 ? 4 3 9 Determine the length of the unknown side. ? 2 4 24 These dodecagons differ by a factor of 6. ? 2 4 24 Sometimes the factor between 2 figures is not obvious and some calculations are necessary. 15 12 18 ?= 8 10 12 To find this missing factor, divide 18 by 12. 15 12 18 ?= 8 10 12 18 divided by 12 = 1.5 The value of the missing factor is 1.5. 15 12 18 1.5 = 8 10 12 This is termed as SSS similarity criterion To calculate a length 15 5x 5 3 4 x3 6 Scale factor 3 15 1 3 18 Scale factor 1/3 12 Find the length of the missing side. 50 30 ? 6 40 8 This looks messy. Let’s translate the two triangles. 50 30 ? 6 40 8 Now “things” are easier to see. 50 30 ? 6 40 8 The common factor between these triangles is 5. 50 30 ? 6 40 8 So the length of the missing side is…? That’s right! It’s ten! 50 30 10 6 40 8 Similar Triangles Triangles are similar if matching angles remain the same size. 100º 30º 50º 100º 30º 50º 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 the angles were 40 increased, the sum 40 would exceed 180 degrees. 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 The other 70 degree angles are congruent. 4 40 70 7070 70 70 •This is termed as AAA similarity criterion 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. You can measure the length of the tree’s shadow. 10 feet 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. 10 feet 6 ft 2 feet The tree must be 30 ft tall. Boy, that’s a tall tree! 10 feet 6 ft 2 feet Similar figures “work” just like equivalent fractions. 30 66 5 11 These numerators and denominators differ by a factor of 6. 30 6 66 6 5 11 Two equivalent fractions are called a proportion. 30 66 5 11 Similar Figures • So, similar figures are two figures that are the same shape and whose sides are proportional. Practice Time! 1) Determine the missing side of the triangle. ? 3 5 4 9 12 1) Determine the missing side of the triangle. 15 3 5 4 9 12 2) Determine the missing side of the triangle. 6 6 36 36 4 ? 2) Determine the missing side of the triangle. 6 6 36 36 4 24 3) Determine the missing sides of the triangle. 39 33 ? ? 8 24 3) Determine the missing sides of the triangle. 39 33 13 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 Show that these triangles are similar 10º 50º 120º 10º 50º 120º …and then… AB & DE are parallel Explain why ABC is similar to CDE 5 A B <CED = <BAC Alternate Angles <EDC = <ABC Alternate Angles <ECD = <ACB Vert Opp Angles 3 C 6 E D ? Triangle ABC is similar to Triangle CDE …and then… Calculate the length of DE AC corresponds to CE Scale Factor = 2 5 A B AB corresponds to DE DE = 2 x AB 3 C DE = 10cm 6 E D ? Summary – Similar shapes • To calculate missing sides, we first of all need the scale factor • We then either multiply or divide by the scale factor • To show that 2 shapes are similar we can either show that all of the sides are connected by the scale factor or show that matching angles are the same THE END!