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2.6 – Similar Triangles
Objectives
 Define similar as it relates to geometric figures and list the key characteristics of similar triangles (congruent angles and proportional sides).
 Dilate geometric figures to determine if they are similar.
 Use correct notation to indicate similarity.
 Calculate the scale factor of triangles by setting up ratios of corresponding sides.
Key Terms
 Dilate – To change the size but not the shape of a geometric figure.
 The dilated figure is similar to the original but is not congruent to it.
 A dilation stretches or shrinks the original figure by a certain scale factor in relation to a point called the center of dilation
 The center of dilation is the point from which the scale factor between the original image points and dilation points is determined.
 In a dilation, segments that go through the center of dilation remain unchanged and segments that do not will become stretched.
 Proportional – Having equal ratios.
 Ratio – A comparison that shows the relative size of one quantity with respect to another.
 The ratio a to b is often written with a colon (a:b) or as a fraction (a/b).
 Scale Factor – The ratio of the lengths of corresponding sides in similar figures.
 In a dilation, this is the factor by which the original figure is multiplied.
 Multiply by a proper fraction (a number smaller than 1) to compress (shrink, or make smaller)
 Multiply by a number larger than 1 to enlarge (grow, get make bigger)
 Whole numbers and decimals work, as long as the value is > 1
 Similar – Having exactly the same shape.
 When figures are similar, corresponding angles are equal (congruent) in measure and corresponding sides are proportional in length.
 The symbol ~ means “is similar to.”
 Notation:
 Similar Triangles – Triangles that have exactly the same shape.
 Their corresponding angles are equal (congruent)
 Their corresponding sides are proportional in length (same ratio).
Notes
Similar Triangles 
 Similar Triangles
 Triangles have equal (congruent) angle measures, but may not have congruent side measures.
 Triangles have proportional side measures
 ALL congruent triangles are similar
 SOME similar triangles are congruent
 The ratios of all three pairs of corresponding sides are equal
 The corresponding angles are equal (congruent)
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 Determining Similarity
 Are the side lengths proportional?
 Do the fractions reduce down to the same value?
 Are all three angle measures equal (congruent)?
 If you answer yes to all of these questions, then you have similar triangles!

Dilating Triangles 
 Dilating Triangles will compress (shrink) or enlarge (grow) them, without changing the shape
 If a triangle gets bigger, the scale factor is > 1 (greater than 1)
 If a triangle gets smaller, the scale factor is < 1 (less than 1, which is a fraction)
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Image Copyright: Apex Learning

Scale Factor 
 To Determine Scale Factor
 Step 1: List ALL of the corresponding sides of a triangle
 If the first triangle gets larger (grows, or enlarges), put the corresponding sides of the larger triangle top of the fraction (numerator) and put the corresponding sides of the smaller triangle on the bottom of the fraction (denominator)
 If the first triangle gets smaller (compresses, or shrinks), put the corresponding sides of the smaller triangle top of the fraction (numerator) and put the corresponding sides of the larger triangle on the bottom of the fraction (denominator)
 Step 2: Reduce the fraction
 Step 3: Check your answer against your knowledge:
 Did the triangles enlarge (grow) or compress (shrink) from the 1st to the 2nd?
 Is the result larger than 1 or smaller than 1?
 If the triangles compress (shrink), the fraction should be less than 1
 If the triangles enlarge (grow), the fraction should be greater than 1

Examples 
 Example of Dilation (Compression)
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 Example of Similar Triangles, NOT Congruent
 All angles measures of the 1st triangle are equal (congruent) to all angle measures on the 2nd triangle

 The ratio for all three triangles is reduced to
 The first triangle is 1/2 the size of the second one
 The dilation has doubled from the first triangle to the 2nd
 So, the scale factor = 2 (because the triangle got larger)
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 Example of Similar Triangles that ARE Congruent
 All angles measures of the 1st triangle are equal (congruent) to all angle measures on the 2nd triangle
 All side lengths of the 1st triangle are congruent to all side lengths on the 2nd triangle (shown with hashmarks)
 The triangle side lengths have a proportional ration of 1:1
 Written as a fraction or just 1.
 The scale factor = 1
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 Example of Finding the Scale Factor
 What is the scale factor of ?
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 The triangles dilate larger (enlarge or grow), so:
 Reduce the fractions:
 They all equal
 Scale Factor = 8

 Lesson Video – Dilation & Scale
 Click icon in the bottom right to view in “Full Screen” mode

Important!
Practice (Apex Study 2.6)
 Click through the animations on Pgs 4, 5, 16
 Practice: Pgs 8, 10, 17, 22, 23
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