Question

madison is given rectangle ( abcd ) with one diagonal ( overline{bd} ).

she wants to prove ( \triangle abdcong\triangle cdb ) using only rigid transformations.

select from the drop - down menus to explain the proof correctly.

madison decides to ( \triangle abd180^{circ} ) about point ( b ) to create ( \triangle abd ). next, she ( ) onal ( overline{bd} ) until point ( b ) from ( \triangle abd ) lines up with point ( d ) from ( \triangle cdb ).

Explanation:

Step1: Recall rigid transformation types

Rigid transformations include translation, reflection, and rotation. Dilation is not a rigid transformation as it changes the size of the figure. Since we are dealing with \(180^{\circ}\) transformation about a point, rotation is the relevant rigid transformation. Rotation of a figure \(180^{\circ}\) about a point changes the orientation of the figure while keeping its shape and size (rigid property).

Step2: Analyze the second transformation

After rotation, the next step is to move the rotated triangle (while keeping its shape and size - rigid property) along the diagonal. Translation is the process of sliding a figure without rotating or resizing it. So, after rotating \(\triangle ABD\) \(180^{\circ}\) about point \(B\) to get \(\triangle A'B'D'\), we translate \(\triangle A'B'D'\) along the diagonal \(\overline{BD}\) until point \(B'\) (from \(\triangle A'B'D'\)) lines up with point \(D\) (from \(\triangle CDB\)).

Answer:

First drop - down: rotate; Second drop - down: translate