Question

which best explains why all equilateral triangles are similar? all equilateral triangles can be mapped onto each other using dilations. all equilateral triangles can be mapped onto each other using rigid transformations. all equilateral triangles can be mapped onto each other using combinations of dilations and rigid transformations. all equilateral triangles are congruent and therefore similar, with side lengths in a 1:1 ratio.

Explanation:

1. Recall the definition of similar figures: Similar figures have the same shape (corresponding angles equal) and their corresponding sides are in proportion. For equilateral triangles, all angles are \(60^\circ\), so corresponding angles are equal. To map one equilateral triangle to another, we can use a dilation (to adjust the size) and then rigid transformations (translation, rotation, reflection) to align them.
2. Analyze Option 1: Dilations alone can change the size but may not align the triangles perfectly without rigid transformations.
3. Analyze Option 2: Rigid transformations (like translation, rotation, reflection) preserve size and shape. But equilateral triangles can have different side lengths, so rigid transformations alone can't map a larger equilateral triangle to a smaller one (or vice versa) since they preserve size.
4. Analyze Option 3: Since equilateral triangles have all angles equal (\(60^\circ\)), we can first use a dilation to make their side lengths proportional (since dilation preserves angle measures) and then use rigid transformations (translation, rotation, reflection) to align them. This combination ensures that any two equilateral triangles can be mapped onto each other, satisfying the similarity condition.
5. Analyze Option 4: Not all equilateral triangles are congruent. Congruent triangles have the same size and shape, but equilateral triangles can have different side lengths (e.g., a triangle with side length 2 and a triangle with side length 3 are equilateral but not congruent). So this statement is incorrect.

Answer:

C. All equilateral triangles can be mapped onto each other using combinations of dilations and rigid transformations.