Question

which of the following transformations carry the isosceles trapezoid onto itself? reflection across $\ell_1$ reflection across $\ell_2$ rotation of $90^{circ}$ counterclockwise rotation of $45^{circ}$ counterclockwise

Explanation:

Step1: Analyze reflection across \(l_1\)

A reflection across \(l_1\) does not carry the isosceles trapezoid onto itself. Because the sides and angles of the trapezoid are not symmetric with respect to \(l_1\).

Step2: Analyze reflection across \(l_2\)

An isosceles trapezoid has a line of symmetry. The line \(l_2\) is the vertical line passing through the mid - points of the non - parallel sides (bases). A reflection across \(l_2\) (the line of symmetry of the isosceles trapezoid) will map each point of the trapezoid to another point on the trapezoid. For any point \(P\) on the trapezoid, if we find its mirror image \(P'\) across \(l_2\), \(P'\) is also on the trapezoid.

Step3: Analyze rotation of \(90^{\circ}\) counterclockwise

A rotation of \(90^{\circ}\) counterclockwise about any point (not the center of symmetry, and an isosceles trapezoid has no center of symmetry for \(90^{\circ}\) rotation) will not map the trapezoid onto itself. The angles and side - length relationships of the trapezoid are not preserved under a \(90^{\circ}\) rotation.

Step4: Analyze rotation of \(45^{\circ}\) counterclockwise

A rotation of \(45^{\circ}\) counterclockwise about any point (an isosceles trapezoid has no center of symmetry for \(45^{\circ}\) rotation) will not map the trapezoid onto itself. The non - parallel sides and the angles of the trapezoid are not invariant under a \(45^{\circ}\) rotation.

Answer:

reflection across \(l_2\)