Question

which of the following transformations carry the given regular polygon onto itself? image of an isosceles triangle with a vertical line l through its apex and base midpoint options: translation, reflection, dilation, none of the choices

Explanation:

Step1: Analyze Translation

A translation moves a figure without rotating or reflecting it. For a regular triangle, translating it would move it to a new position, not onto itself. So translation is not correct.

Step2: Analyze Reflection

The given triangle is isosceles (regular, so equilateral, which is a type of isosceles) with a line of symmetry \( L \). Reflecting the triangle over line \( L \) will map each point on one side of \( L \) to the corresponding point on the other side, carrying the triangle onto itself.

Step3: Analyze Dilation

Dilation changes the size of the figure (unless the scale factor is 1, but dilation typically refers to non - identity scale factors in this context). A dilation with a scale factor not equal to 1 will make the triangle larger or smaller, not map it onto itself. A dilation with scale factor 1 is a trivial transformation, but it's not the typical dilation considered here. So dilation is not correct.

Step4: Analyze "None of the choices"

Since reflection works, this option is incorrect.

Answer:

Reflection (the option: Reflection)