Question

triangle tuv and triangle ijk are drawn below. if \\(\overline{tv} \cong \overline{ik}\\), \\(\angle v \cong \angle k\\), and \\(\angle t \cong \angle i\\), use the dropdown boxes below to determine a sequence of two transformations that maps triangle tuv onto triangle ijk. then use a congruence statement to explain why this is possible. image of triangles tuv and ijk rotate triangle tuv counterclockwise about point dropdown until dropdown has the same slope as dropdown. then dropdown triangle tuv tryreset

Explanation:

Step1: Determine the rotation center

Since we want to map \(TV\) to \(IK\) (because \(\overline{TV}\cong\overline{IK}\)), we rotate triangle \(TUV\) counter - clockwise about point \(T\) (or \(I\), but since we start with \(TUV\), we choose \(T\)). We rotate until \(\overline{TV}\) has the same slope as \(\overline{IK}\).

Step2: Determine the second transformation

After rotation, we need to translate the rotated triangle \(T'U'V'\) (the image of \(TUV\) after rotation) to map it onto \(IJK\).

Step3: Congruence statement

We know that \(\angle V\cong\angle K\), \(\angle T\cong\angle I\) and \(\overline{TV}\cong\overline{IK}\). By the \(AAS\) (Angle - Angle - Side) congruence criterion, \(\triangle TUV\cong\triangle IJK\). So a rotation (to align the sides and angles) followed by a translation (to move the triangle to the exact position) will map \(\triangle TUV\) onto \(\triangle IJK\).

Answer:

Rotate triangle \(TUV\) counter - clockwise about point \(T\) until \(\overline{TV}\) has the same slope as \(\overline{IK}\). Then translate triangle \(T'U'V'\). The congruence statement is \(\triangle TUV\cong\triangle IJK\) by \(AAS\) (since \(\angle V\cong\angle K\), \(\angle T\cong\angle I\) and \(\overline{TV}\cong\overline{IK}\)), so the transformation (rotation + translation) is possible.