Geometry Chapter 7 Test Flashcards

Ratios and Proportions

• Ratio: A comparison of two quantities through division.
• Proportion: An equation that asserts the equality between two ratios.
• Extended Ratio: A ratio that compares three or more quantities.

Triangle Measurements

• Triangle Ratios: For triangle sides in a ratio of 2:2:3 with a perimeter of 392, use 2x + 2x + 3x = 392 to find side lengths.
• Perimeter Without Given Value: Set the ratios of triangle sides to equal 180 if perimeter is not provided.

Scale Models and Dilation

• Scale Model: An object or drawing that maintains proportional lengths relative to the object it represents.
• Dilation: A transformation that alters the size of an object without changing its shape.
• Scale Factor: Represents the ratio of an image's length to the original object's length, e.g., 3:25 simplified to 1:100.

Similar Polygons

• Similar Polygons: They share the same shape, though sizes may vary.
• Rule #1: In similar polygons, all corresponding angles are congruent.
• Rule #2: All corresponding sides within similar polygons remain proportional.

Perimeter and Triangle Theorems

• Proportional Perimeters: If two figures are similar, their perimeters correspond to the scale factor.
• Proportional Parts of Triangles: Non-parallel transversals intersecting parallel lines can create similar triangles.
• Side Splitter Theorem: A line parallel to one side of a triangle divides the other two sides proportionally (AB/BC = AE/ED).
• Converse of Side Splitter: If segments are proportional, the sides are parallel.
• Triangle Midsegment Theorem: Midsegment is parallel to one triangle side and half its length.

Advanced Theorems

• Proportional Perimeter Theorem: If figures F and G are similar, proportions of corresponding sides and perimeters are equal.
• Corresponding Altitudes and Medians Theorems: Corresponding segments remain proportional for similar figures.

Dilation Properties

• Center of Dilation: The fixed point used as a reference when performing dilations.
• Types of Dilation:
  • Enlargement: Dilation with a scale factor greater than 1.
  • Reduction: Dilation with a scale factor between 0 and 1.

Finding Scale Factors

• Scale Factor of Enlargement: Calculate the ratio of distances between two points on the smaller image to the larger image.
• Scale Factor of Reduction: Compare distances from the larger image to the smaller image.

Transformations

• Reflections: images are symmetrically opposite along a segment or point.
  • X-Axis Reflection: Invert the y-coordinate. Transformation results in (x, -y).
  • Y-Axis Reflection: Invert the x-coordinate. Transformation results in (-x, y).
• Translations: Shifts a point by adding or subtracting from the coordinates. Represents movement along a vector (x + a, y + b).
• Rotations:
  • 90°: (x, y) becomes (-y, x).
  • 180°: (x, y) becomes (-x, -y).
  • 270°: (x, y) becomes (y, -x).