Question

1. in the diagram shown, \\( \triangle lmn \\) is the image of \\( \triangle ijk \\) after a clockwise rotation about point \\( c \\) by \\( 70^{\circ} \\). answer the following questions.

(a) if an arc was drawn through point \\( i \\) with a center at point \\( c \\), what other marked point on the diagram would it pass through?
(b) the distance from point \\( c \\) to point \\( m \\) on the diagram is equal to the distance from point \\( c \\) to what other marked point?
(c) if \\( m \angle jki = 29^{\circ} \\), name another angle whose measure is \\( 29^{\circ} \\). mark on the diagram.
(d) draw two rays starting at point \\( c \\) such that the angle formed is \\( 70^{\circ} \\). name the angle below and mark its measure in the diagram.

Explanation:

Step1: Recall rotation properties

Rotation is a transformation that preserves distances from the center of rotation and congruence of figures.

Step2: Solve part (a)

Since \(\triangle LMN\) is the image of \(\triangle IJK\) after a rotation about \(C\), the distance from \(C\) to \(I\) is equal to the distance from \(C\) to \(L\) (rotation preserves distances from the center). So an arc centered at \(C\) passing through \(I\) will pass through \(L\).

Step3: Solve part (b)

Rotation preserves distances from the center of rotation. Point \(M\) and point \(J\) are corresponding points in the pre - image (\(\triangle IJK\)) and image (\(\triangle LMN\)) of the rotation about \(C\). So \(CM = CJ\).

Step4: Solve part (c)

Rotation preserves angle measures. \(\angle JKI\) and \(\angle LNM\) are corresponding angles in the congruent triangles \(\triangle IJK\) and \(\triangle LMN\) (because of rotation). So \(m\angle LNM=29^{\circ}\).

Step5: Solve part (d)

The angle of rotation is \(70^{\circ}\). If we consider the rotation of a point \(P\) around \(C\) to \(P'\), the angle \(\angle PCP'\) is the angle of rotation. For example, if we consider the rotation of \(I\) to \(L\) (or \(J\) to \(M\) or \(K\) to \(N\)), \(\angle ICL = 70^{\circ}\) (or \(\angle JCM=70^{\circ}\) or \(\angle KCN = 70^{\circ}\))

Answer:

(a) Point \(L\)
(b) Point \(J\)
(c) \(\angle LNM\)
(d) \(\angle ICL\) (or \(\angle JCM\) or \(\angle KCN\)) with measure \(70^{\circ}\)