Question

the transformation t(3,8) o r(90°,a) maps △abc to △def. is △abc congruent to △def? what kind of transformation is t(3,8) o r(90°,a)?
triangle abc congruent to △def because t(3,8) o r(90°,a) is

Explanation:

Step1: Understand congruence and transformations

Congruent triangles have the same shape and size. Transformations like rotation (\(R(90^{\circ},A)\)) and translation (\(T(3,8)\)) preserve shape and size.

• A rotation \(R(90^{\circ},A)\) rotates a figure \(90^{\circ}\) around point \(A\). It is a rigid transformation, so the shape and size of \(\triangle ABC\) remain unchanged after rotation.
• A translation \(T(3,8)\) moves every point of a figure \(3\) units in the \(x -\) direction and \(8\) units in the \(y -\) direction. It is also a rigid transformation, so the shape and size of the rotated figure (after \(R(90^{\circ},A)\)) remain unchanged after translation.

Step2: Determine the type of transformation

The composition \(T(3,8)\circ R(90^{\circ},A)\) is a composition of two rigid transformations.

• A rigid transformation (also called an isometry) is a transformation that preserves distances and angles. Rotations and translations are rigid transformations.
• When we compose two rigid transformations (first rotate \(R(90^{\circ},A)\) then translate \(T(3,8)\)), the resulting transformation \(T(3,8)\circ R(90^{\circ},A)\) is also a rigid transformation.

Answer:

\(\triangle ABC\) is congruent to \(\triangle DEF\) because \(T(3,8)\circ R(90^{\circ},A)\) is a composition of rigid transformations (a rotation and a translation). Rigid transformations preserve the shape and size of a figure. So, the two triangles \(\triangle ABC\) and \(\triangle DEF\) have the same shape and size (by the definition of congruent triangles: two triangles are congruent if there is a sequence of rigid transformations that maps one triangle to the other).