Question

a square is shown below.
line ( m ) intersects two vertices.
line ( n ) bisects each side it passes through.
point ( k ) is the center of the square.
which transformation(s) must map the square exactly onto itself? choose all that apply.
reflection across line ( m )
counterclockwise rotation about ( k ) by ( 270^{circ} )
reflection across line ( n )
clockwise rotation about ( k ) by ( 180^{circ} )
none of the above

Explanation:

Step1: Analyze reflection across line \(m\)

A square has symmetry about its diagonals. Reflection across a diagonal (line \(m\)) maps the square onto itself.

Step2: Analyze counter - clockwise rotation about \(K\) by \(270^{\circ}\)

A square has rotational symmetry of order \(4\). The angle of rotational symmetry is \(\frac{360^{\circ}}{4} = 90^{\circ}\). A \(270^{\circ}\) counter - clockwise rotation (\(270\div90 = 3\) full "steps" of \(90^{\circ}\) rotation) about the center \(K\) maps the square onto itself.

Step3: Analyze reflection across line \(n\)

A square has symmetry about the lines that bisect its sides (line \(n\)). Reflection across such a line maps the square onto itself.

Step4: Analyze clockwise rotation about \(K\) by \(180^{\circ}\)

Since \(180\div90=2\), a \(180^{\circ}\) clockwise rotation about the center \(K\) (which is equivalent to two \(90^{\circ}\) rotations) maps the square onto itself.

Answer:

Reflection across line \(m\), Counterclockwise rotation about \(K\) by \(270^{\circ}\), Reflection across line \(n\), Clockwise rotation about \(K\) by \(180^{\circ}\)