S - Dilations and Similarity Module Overview
Dilations and Similarity Module Overview
How tall is that building? How wide is that river? How deep is that canyon? These are all questions that would be difficult to answer with a ruler or tape measure even if you had the time to try, but using geometry, they can be answered from a distance. No need to climb, cross or hike. With a simple drawing and some calculations, you can answer these questions and more! Architects use these concepts to plan construction projects. Manufacturers use them to build scale models and design new planes, cars, phones and just about everything else. In this unit, you will learn about the special relationships between similar objects, as well as the connection between dilation and similarity. You will then be able to use dilation and similarity to measure objects, discuss scale, and solve other mathematical puzzles.
Special Note: You will need a compass and ruler for this module.
Essential Questions
- What is a dilation and how does this transformation affect a figure in the coordinate plane?
- How does the location of the center of dilation determine the effect the dilation has on a given line?
- Why is it important to understand proportionality when studying similarity?
- How can I know two figures are similar?
- How do I know which method to use to prove two triangles are similar?
- What transformations result in similar figures?
- What strategies can I use to determine missing side lengths and areas of similar figures
Key Terms
The following key terms will help you understand the content in this module.
Angle - Angles are created by two distinct rays that share a common endpoint (also known as a vertex). ∠ ABC or ∠ B denote angles with vertex B.
Bisector - A bisector divides a segment or angle into two equal parts.
Congruent - Having the same size, shape and measure. Two figures are congruent if all of their corresponding sides and angle measures are equal.
Congruent Figures - Figures that have the same size and shape.
Corresponding Angles - Angles that have the same relative positions in geometric figures.
Corresponding Sides - Sides that have the same relative positions in geometric figures.
Dilation - Transformation that changes the size of a figure, but not the shape.
Line - One of the basic undefined terms of geometry. Traditionally thought of as a set of points that has no thickness but its length goes on forever in two opposite directions. ↔AB denotes a line that passes through point A and B.
Line Segment or Segment - The part of a line between two points on the line. ¯AB denotes a line segment between the points A and B.
Midsegment - A line segment whose endpoints are the midpoint of two sides of a triangle is called a midsegment of a triangle.
Parallel Lines - Two lines are parallel if they lie in the same plane and they do not intersect. Parallel lines have the same slope.
Point - One of the basic undefined terms of geometry. Traditionally thought of as having no length, width, or thickness, and often a dot is used to represent it.
Proportion - An equation which states that two ratios are equal.
Ratio - Comparison of two quantities by division and may be written as r/s, r:s, or "r to s."
Ray - A ray begins at a point and goes on forever in one direction.
Reflection - A transformation that "flips" a figure over a line of reflection.
Rotation - A transformation that turns a figure about a fixed point through a given angle and a given direction.
Scale Factor - The ratio of any two corresponding lengths of the sides of two similar figures.
Similar Figures - Figures that have the same shape but not necessarily the same size.
Transformation - The mapping, or movement, of all the points of a figure in a plane according to a common operation.
Translation - A transformation that "slides" each point of a figure the same distance in the same direction.
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