Representing Proportional Relationships

Properties of Dilations8.G.4Essential Question?How do you describe the properties of dilations?

Common Core Standard:8.G Understand congruence and similarity using physical models, transparencies, or geometry software.4. Understand that a two-dimensional figure is similar to another if the second can be obtained from the first by a sequence of rotations, reflections, translations, and dilations; given two similar two-dimensional figures, describe a sequence that exhibits the similarity between them.Objectives:To describe the properties of dilation and their effect on the similarity and orientation of figures.Curriculum VocabularyCenter of Dilation (centro de dilatacin):The point of intersection of lines through each pair of corresponding vertices in a dilation.Dilation (dilatacin):A transformation that moves each point along the ray through the point emanating from a fixed center, and multiplies distances from the center by a common scale factor.Ratio (razn):A comparison of two quantities by division.Scale (escala):The ratio between two sets of measurements.Curriculum VocabularyEnlargement (agrandamiento):An increase in the size of all dimensions in the same proportions.Reduction (reduccin):A decrease in the size of all dimensions in the same proportions.Scale Factor (factor de escala):The ratio used to enlarge or reduce similar figures.Similar (similar / semejantes):Figures with the same shape but not necessarily the same size.Ratios of corresponding sides are proportionalMeasures of corresponding angles are equalProperties of DilationsWe often see a scale model or a toy that replicates a real object.A replica is a transformation called a DILATION.Unlike the other transformations you have studied: Translations Rotations ReflectionsDilations change the size (but not the shape)of a figure.orientation is preserved.Every dilation has a fixed point called theCENTER OF DILATIONlocated where the lines connecting corresponding parts of figures intersect.DilationsLets look at some dilations:

The dilation of this tiger made it bigger.IMAGEPREIMAGEThe dilation of this smiley made it smaller.

In all dilations, the image and pre-image look exactly the same. The only thing that changes is the size.DilationsIn this drawing, RST is a dilation of RST.Point C is the center of dilation.Lets work together to complete the worksheet.

PREIMAGEIMAGECENTER OF DILATIONFill in the following information:Center of Dilation: ___________Image: ___________Preimage: ___________Use a ruler to measure each of the line segments to the nearest centimeter:

Now use your answers to find the following RATIOS:(SIMPLIFY EACH FRACTION!)

Use a protractor to measure each of the angles:

Based on the information you found in PART 3, what can you conclude about the ratios of corresponding sides?

Based on the information you found in PART 4, what can you conclude about the measures of corresponding angles?

Each line segment in the image is ______ times its corresponding line segment in the preimage.

Recall that the ratio used to enlarge or reduce similar figures is called the __________________.

The dilation of RST to RST uses a ______________________ of _______.DilationsNow lets examine a dilation on a coordinate plane.Which is the image andwhich is the preimage?What is the center of dilation?Fill out the table:

VertexxyVertexxyAABBCCDD

VertexxyVertexxyA12.5A25B21B42C22C44D22D44Ratios for x-coordinatesRatios for y-coordinatesA & AA & AB & BB & BC & CC & CD & DD & D

In our first example, RST is BIGGER than RST.In our second example, ABCD is SMALLER than ABCD.When the image is BIGGER than the preimage it is anENLARGEMENTWhen the image is SMALLER than the preimage it is aREDUCTION

PREIMAGEIMAGECENTER OF DILATION

PREIMAGEIMAGEENLARGEMENTREDUCTIONSCALE FACTORWe already know thatSCALE FACTORis the ratio used to enlarge or reduce similar figures.With an ENLARGEMENT, the SCALE FACTOR is ALWAYS a number GREATER THAN 1.With a REDUCTION, the SCALE FACTOR isALWAYS a number BETWEEN 0 and 1.SCALE FACTORFind the SCALE FACTOR of the dilation:

Is the dilation an enlargement or reduction?

How do you know?