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:

Step1: Properties of equilateral triangles

All equilateral triangles have equal angles (\(60^{\circ}\) each).

Step2: Similarity and transformations

Rigid transformations (translations, rotations, reflections) preserve shape and size ratios. Dilations change size but preserve shape.
Since equilateral triangles have equal - angled measures (AA similarity criterion: if two angles of one triangle are equal to two angles of another triangle, the triangles are similar), and we can use a dilation (to adjust size) and rigid transformations (to adjust position/orientation) to map one equilateral triangle to another.

• Option 1: Dilations alone may not account for position/orientation.
• Option 2: Rigid transformations alone cannot account for size differences (if triangles are of different sizes).
• Option 4: Not all equilateral triangles are congruent (they can have different side lengths).

So, the correct option is “All equilateral triangles can be mapped onto each other using combinations of dilations and rigid transformations.”

Answer:

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