Question

triangle \\( \triangle a ^ { prime } b ^ { prime } c ^ { prime } \\) is the result of translating \\( \triangle a b c \\) by 6 units to the right and 2 units down.select all of the correct statements about the unchanged properties of \\( \triangle a b c \\) and \\( \triangle a ^ { prime } b ^ { prime } c ^ { prime } \\).choose all answers that apply\\( \square \\) \\( \angle a \\) and \\( \angle a ^ { prime } \\) have the same measures.\\( \square \\) \\( b \\) and \\( b ^ { prime } \\) have the same coordinates.\\( \square \\) the areas of \\( \triangle a b c \\) and \\( \triangle a ^ { prime } b ^ { prime } c ^ { prime } \\) are the same.\\( \square \\) none of the above

Explanation:

Step1: Properties of translation

Translation is a rigid transformation. Rigid transformations preserve angles and areas. So, \(\angle A\) and \(\angle A'\) (corresponding angles of the two triangles) have the same measure because translation does not change the shape and size of the figure, which affects the angle measures. Also, the area of \(\triangle ABC\) and \(\triangle A'B'C'\) is the same since translation is a congruence transformation (the two triangles are congruent, and congruent triangles have equal areas).

Step2: Coordinates of points

When we translate a point \((x,y)\) by \(h\) units to the right and \(k\) units down, the new coordinates are \((x + h,y - k)\). So, the coordinates of \(B\) and \(B'\) are different. For example, if \(B=(x_{B},y_{B})\), then \(B'=(x_{B}+6,y_{B}-2)\)

Answer:

A. \(\angle A\) and \(\angle A'\) have the same measures; C. The areas of \(\triangle ABC\) and \(\triangle A'B'C'\) are the same.