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:

Similar triangles have equal corresponding angles and proportional corresponding sides. All equilateral triangles have \(60^{\circ}\) angles. A dilation (a non - rigid transformation that changes the size but not the shape) can be used to map one equilateral triangle to another. Rigid transformations (like translations, rotations, reflections) can also be used to re - position the triangle. Combining dilations (to adjust size) and rigid transformations (to adjust position) can map any equilateral triangle to another. Rigid transformations alone (second option) can't account for size differences if triangles are not congruent. First option (only rigid transformations) is incorrect for non - congruent equilateral triangles. Fourth option is wrong because not all equilateral triangles are congruent (only when side lengths are equal).

Answer:

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