Math Problem Statement
Describe a sequence of transformations that moves the shaded pre-image onto the unshaded image to show that the figures are similar.
Solution
The shaded and unshaded triangles are similar, and we can demonstrate this by describing a sequence of transformations. Here’s the process to move the shaded triangle onto the unshaded triangle :
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Translation: Move the shaded triangle such that point aligns with point . This can be done by translating all the vertices by the vector . This translation ensures that maps to , while the relative positions of points and are preserved.
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Dilation: After the translation, dilate the triangle with center and a scale factor. The scale factor can be found by comparing the lengths of corresponding sides in the triangles: This dilation will enlarge the smaller triangle to match the size of , making them congruent.
Since these two transformations (translation and dilation) map the shaded triangle exactly onto the unshaded triangle, they are similar by definition. Similarity is established when figures can be mapped onto each other using a sequence of rigid transformations (like translation) and dilations.
Would you like more details on any step, or do you have any further questions?
Here are five related questions:
- What properties define similar triangles?
- How do you determine the scale factor for dilation in similar figures?
- Why do translations not affect the size or shape of a figure?
- What is the difference between congruence and similarity in geometry?
- Can a triangle be mapped to another triangle using just a reflection? Why or why not?
Tip: In similarity transformations, angles remain unchanged, but the lengths of corresponding sides change by a constant ratio.
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Math Problem Analysis
Mathematical Concepts
Geometry
Transformations
Similarity
Translation
Dilation
Formulas
Scale factor = (length of A'B') / (length of AB)
Scale factor = (length of A'C') / (length of AC)
Theorems
Similarity Theorem
Properties of Transformations
Suitable Grade Level
Grades 8-10
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