Question

triangle abc is an equilateral triangle. point p is equidistant from all three vertices. which transformations carry triangle abc onto itself? select all that apply. a reflection across line ap a reflection across line bp a reflection across line ac a reflection across line cp

Explanation:

Step1: Recall properties of equilateral triangle

An equilateral triangle has 3 lines of symmetry. The lines connecting a vertex to the mid - point of the opposite side are lines of symmetry. Also, if a point \(P\) is equidistant from all three vertices of an equilateral triangle, \(P\) is the circum - center and the in - center. The lines \(AP\), \(BP\), and \(CP\) are lines of symmetry.

Step2: Analyze reflections

A reflection across a line of symmetry of a figure maps the figure onto itself. For an equilateral triangle \(ABC\) with a point \(P\) (circum - center), reflections across \(AP\), \(BP\), \(CP\) will carry the triangle onto itself. Also, a reflection across the line joining the mid - point of a side and the opposite vertex (e.g., the perpendicular bisector of \(AC\)) is a line of symmetry and will map the triangle onto itself.

Answer:

a reflection across line \(AP\), a reflection across line \(BP\), a reflection across line \(AC\), a reflection across line \(CP\)