Question

an equilateral triangle is shown below.
line c passes through a vertex and bisects a side.
line d bisects each side it passes through.
point p is the center of the triangle.
which transformation(s) must map the triangle exactly onto itself? choose all that apply.
reflection across line d
reflection across line c
clockwise rotation about p by 60°
counterclockwise rotation about p by 144°
none of the above

Explanation:

Step1: Properties of an equilateral triangle

An equilateral triangle has rotational symmetry of order 3 (rotational angle \(120^{\circ}\)) and reflection symmetry across its three angle - bisectors (which are also the medians and altitudes).

Step2: Analyze each transformation

• Reflection across line \(c\):

Since line \(c\) is an angle - bisector (and thus a line of symmetry) of the equilateral triangle, reflection across line \(c\) will map the triangle onto itself.

• Reflection across line \(d\):

Since line \(d\) is an angle - bisector (and thus a line of symmetry) of the equilateral triangle, reflection across line \(d\) will map the triangle onto itself.

• Clockwise rotation about \(P\) by \(60^{\circ}\):

The center \(P\) of an equilateral triangle has rotational symmetry of \(120^{\circ}\). A \(60^{\circ}\) rotation about \(P\) will not map the triangle onto itself.

• Counter - clockwise rotation about \(P\) by \(144^{\circ}\):

Since \(144^{\circ}\) is not a multiple of \(120^{\circ}\), a counter - clockwise rotation about \(P\) by \(144^{\circ}\) will not map the triangle onto itself.

Answer:

Reflection across line \(c\), Reflection across line \(d\)