Daniel Cohen & Jacob Singer

8.1 = Dilations and Scale Factors

8.2 = Similar Polygons

8.3 = Triangle Similarity

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 )

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.)

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)

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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Dilations and Scale Factors Quiz

Polygon Similarity Quiz