Geometry Honors Chapter 6 Notes

Chapter 6: Similarity

6.1: Dilations

  • Definition of Dilation:

    • A dilation is a transformation that enlarges or reduces a figure proportionally with respect to a center point and a scale factor.

  • Steps for Performing a Dilation:

    1. Draw rays from the center point (e.g., point D) through each vertex of the figure (e.g., A, B, and C).
    2. For each vertex, locate the image (e.g., A′) such that the distance from the center to the image equals the scale factor times the distance from the center to the original vertex.
    3. Draw the new figure using the images of the original vertices.

  • Notation:

    • The original figure is called the pre-image (e.g., ∆ABC), and the transformed figure is called the image (e.g., ∆A′B′C′).

  • Scale Factor (k):

    • Can be greater than 1 (enlargement) or less than 1 (reduction).
    • Abbreviated as the symbol k.

  • Coordinate Dilation Example:

    • If point A has coordinates (3,0) and is dilated by a scale factor of 2 centered at the origin, the new coordinates can be calculated as:
    • A′(x,y) = (k * x, k * y) where k is the dilation factor.
    • Therefore, A′(2 * 3, 2 * 0) = (6,0).

  • General Dilation Formula:

    • For any point (a,b) under dilation centered at the origin with scale factor k:
    • New coordinates = (k * a, k * b)
6.2: Similar Polygons

  • Definition of Similar Polygons:

    • Similar polygons are figures with the same shape but not necessarily the same size.
    • Their corresponding angles are congruent, and their corresponding side lengths are in proportion.

  • Scale Factor for Similar Polygons:

    • The ratio of the lengths of corresponding sides of two similar polygons.

  • Important Properties:

    • If two polygons are similar:
    • Corresponding angles are congruent.
    • Corresponding sides are proportional.
    • If corresponding angles are congruent and corresponding sides are proportional, the polygons are similar.

  • Example of Similarity Statement:

    • If ΔNQP ~ ΔRST, then angles N, Q, P correspond to angles R, S, T, respectively.
6.3: Similar Triangles

  • Triangle Similarity Criteria:

    • Angle-Angle (AA) Criterion:
    • If two angles of one triangle are congruent to two angles of another triangle, the triangles are similar.
    • Side-Side-Side (SSS) Criterion:
    • If the corresponding sides of two triangles are proportional, the triangles are similar.
    • Side-Angle-Side (SAS) Criterion:
    • If two sides of one triangle are proportional to two sides of another triangle, and the included angles are congruent, the triangles are similar.

  • Example Problem:

    • For triangles with corresponding side lengths AB, BC, and corresponding side lengths DE, EF, identify if triangles are similar and find the scale factor.
6.4: Proportional Lengths

  • Triangle Proportionality Theorem:

    • If a line parallel to one side of a triangle intersects the other two sides, it divides these sides proportionally.
    • If a ray bisects an angle of a triangle, it divides the opposite side into segments proportional to the other two sides.

  • Example Statement:

    • AD/DC = AB/AC for triangle properties.

  • Example Application:

    • If parallel lines intersect two transversals, the segments created are proportional, and one can write statements like KL/LM = XY/XZ to indicate this relationship.