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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:  \Delta ABC \sim \Delta DEF
  • 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)

Geo A 2.06 - Similar Triangles 08

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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!

Geo A 2.06 - Similar Triangles 04

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)
Geo A 2.06 - Similar Triangles 06

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Geo A 2.06 - Similar Triangles 07

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dilation

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

Geo A 2.06 - Similar Triangles 05

Examples
  • Example of Dilation (Compression)

Geo A 2.06 - 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
    •  \frac{8}{16}=\frac{10}{20}=\frac{6}{12}
      • The ratio for all three triangles is reduced to \frac{1}{2}
    • 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)

Geo A 2.06 - Similar Triangles 01

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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 \frac{1}{1} or just 1.
      • The scale factor = 1

Geo A 2.06 - Similar Triangles 03

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  • Example of Finding the Scale Factor
    • What is the scale factor of \Delta ABC \sim \Delta DEF?
Geo A 2.06 - Similar Triangles 06

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    • The triangles dilate larger (enlarge or grow), so:
      • \frac{Large}{Small}
      • \frac{\overline{EF}}{\overline{BC}}=\frac{\overline{AB}}{\overline{DE}}=\frac{\overline{DF}}{\overline{AC}}
      • \frac{48}{6}=\frac{48}{6}=\frac{32}{4}
    • Reduce the fractions:
      • They all equal \frac{8}{1}=8
    • Scale Factor = 8
  • Lesson Video – Dilation & Scale
    • Click icon in the bottom right to view in “Full Screen” mode

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