daniel cohen & jacob singer. 8.1 = dilations and scale factors 8.2 = similar polygons 8.3 =...
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Daniel Cohen & Jacob Singer
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8.1 = Dilations and Scale Factors
8.2 = Similar Polygons
8.3 = Triangle Similarity
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Dilations and Scale Factors
Dilation = A transformation that is not rigid. Preserves the shape of an object, but the size may vary. (Example: Your eyes will dilate to adjust to brightness).
Dilations can be found on a coordinate plane by multiplying the x and y coordinates of a point by the same number n. D(x, y) = (nx, ny)
The number n called a scale factor.
Types of Dilations: Contraction = The sixe of a figure is reduced by the dilation. ( l n l < 1 )
Expansion = The size of the figure is enlarged by the dilation. ( l n l > 1 )
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Similar Polygons
Similar Figures = Two figures are similar if and only if one is congruent to the image of the other by a dilation.
Proportional = When the ratios of corresponding sides of two polygons are equal
Proportion = The statement of the equality of two ratios.
Polygon Similarity Postulate = Two polygons can only be similar if each pair of corresponding angles are congruent, and each pair of corresponding sides are proportional. (Hint – The letters of the vertices of the polygons must be written in corresponding order.)
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When working with similar figures, it is often helpful to know the following:
Properties of Proportions:
Cross-Multiplication: If a / b = c / d, then ad = bc (d ≠ 0)
Reciprocal: If a / b = c / d, then b / a = d / c (a, b, c, and d ≠ 0)
Exchange: If a / b = c / d, then a / c = b / d (a, b, c, and d ≠ 0)
“Add One”: If a / b = c / d, then (a + b) / b = (c + d) / d (b and d ≠ 0)
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Triangle Similarity
Here are some helpful shortcuts for determining triangle similarity! Yayyyy!
AA (Angle – Angle) Similarity Postulate: If two angles of a triangle are congruent to two angles of another triangle, then the triangles are similar.
SSS (Side – Side – Side) Similarity Theorem: If three sides of one triangle are proportional to the three sides of another triangle, then the triangles are similar.
SAS (Side – Angle – Side) Similarity Theorem: If two sides of one triangle are proportional to two sides of another triangle and their included angles are congruent, then the triangles are similar.
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Click Here to go to Polygon Similarity land!
Click Here for fun proportion information!
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Click Here to go to Fun Triangle Similarity World!
Dilations and Scale Factors Quiz
Polygon Similarity Quiz