02.05 Similar Figures
Similar Figures: In order for figures to be similar, two things have to be true.
- The first thing is that they have to have shapes that are similar. - The second thing is that the size has to be proportional, by the same scale factor.
Characteristics of Similar Figures
- angles correspond to one another and are congruent - sides correspond to one another and are proportional - sides all have the same proportional relationship
Scale factor: Meaning
A ratio of two corresponding lengths that determines the change in size from a pre-image to an image.
Scale factor
A scale factor that is a number greater than 1 represents a dilation that increases the size of the pre-image. A scale factor that is less than 1 represents a decrease in the size of the pre-image.
Dilation: Meaning
A type of transformation that changes the size but not the shape of a figure.
Similar Versus Congruent
Angles: Similar ∼ Same angle measurements Congruent ≅ Same angle measurements Sides: Similar ∼ Can be the same or different sizes Congruent ≅ Exact same size Transformations Similar ∼ translations, reflections, rotations, or dilations Congruent ≅ translations, rotations, or reflections (not dilations) *** Similar figures can be different sizes.
Similar Versus Congruent: Note!
Congruent figures also qualify as similar. They have the same angle measurements and are the same size. When figures are marked as congruent, you can also assume that they are similar.
Dilation
Dilations are not rigid transformations because the size does not stay the same.
Similar
Having the same shape and angle measure but different sizes; expressed with the symbol ∼. Similar figures are objects that have the same shape and same angle measurements, but different sizes. One is larger than the other. They look a lot alike, but they are not exactly alike; they are similar.
Finding Scale Factor
To find the scale factor, choose two sides that are corresponding or in the same position on the triangle. Set up a ratio with the image measurement on top and the pre-image measurement in the denominator. Put the value of the image in the numerator and the pre-image in the denominator. Simplify if possible.
Putting It Together
Transformations can be combined to create a sequence. Any combination of the movements can create a new image.
Scale factor: Coordinate plane
Use the vertices to find the scale factor, as long as the coordinates are in corresponding location.
Sequences of Transformations
You have learned about four types of transformations: translations (slide), rotations (spin), reflections (flip), and dilations (change in size). Remember that translations, rotations, and reflections create congruent figures. Dilations create similar figures.