Final Answer:
The transformation that will map an isosceles trapezoid onto itself is 3 - Reflection across a line joining the midpoints of the nonparallel sides.
Step-by-step explanation:
A reflection across a line joining the midpoints of the nonparallel sides will preserve the isosceles trapezoid's symmetry and maintain its overall shape. Let's denote the isosceles trapezoid's vertices as A, B, C, and D, where AB and CD are the parallel sides. The midpoints of the nonparallel sides are denoted as E (midpoint of AD) and F (midpoint of BC).
To understand why the reflection across the line EF is the correct transformation, consider the following:
Each point on one side of EF has a corresponding point equidistant on the other side, preserving the trapezoid's symmetry.
The distance between a vertex and the midpoint of the opposite side remains the same after reflection, ensuring that the shape is unchanged.
Therefore, a reflection across the line joining the midpoints of the nonparallel sides is the appropriate transformation to map the isosceles trapezoid onto itself. This choice guarantees that corresponding points maintain the same relative positions, and the trapezoid's internal angles and side lengths remain invariant.