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.

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 ABC\triangle ABC onto the unshaded triangle ABC\triangle A'B'C':

  1. Translation: Move the shaded triangle ABC\triangle ABC such that point AA aligns with point AA'. This can be done by translating all the vertices by the vector AA\overrightarrow{AA'}. This translation ensures that AA maps to AA', while the relative positions of points BB and CC are preserved.

  2. Dilation: After the translation, dilate the triangle with center AA' and a scale factor. The scale factor can be found by comparing the lengths of corresponding sides in the triangles: Scale factor=length of ABlength of AB=length of AClength of AC\text{Scale factor} = \frac{\text{length of } A'B'}{\text{length of } AB} = \frac{\text{length of } A'C'}{\text{length of } AC} This dilation will enlarge the smaller triangle ABC\triangle ABC to match the size of ABC\triangle A'B'C', 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:

  1. What properties define similar triangles?
  2. How do you determine the scale factor for dilation in similar figures?
  3. Why do translations not affect the size or shape of a figure?
  4. What is the difference between congruence and similarity in geometry?
  5. 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