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:

\\( \angle a \\) and \\( \angle a ^ { prime } \\) have the same measures.

\\( b \\) and \\( b ^ { prime } \\) have the same coordinates.

the areas of \\( \triangle a b c \\) and \\( \triangle a ^ { prime } b ^ { prime } c ^ { prime } \\) are the same.

none of the above

Explanation:

• Translation in geometry: A translation is a rigid transformation. Rigid transformations preserve the shape and size of a figure. This means that corresponding angles and side - lengths remain equal.
• Angle measure: Since \(\triangle A'B'C'\) is a translation of \(\triangle ABC\), \(\angle A\) and \(\angle A'\) are corresponding angles. In a rigid transformation (translation), corresponding angles are congruent, so \(m\angle A=m\angle A'\).
• Coordinates: When we translate a point \((x,y)\) by \(a\) units to the right and \(b\) units down, the new coordinates are \((x + a,y - b)\). For point \(B\) (original coordinates \((3,1)\)), after translation by \(6\) units to the right and \(2\) units down, the coordinates of \(B'\) are \((3+6,1 - 2)=(9,-1)\). So \(B\) and \(B'\) do not have the same coordinates.
• Area: Because translation is a rigid transformation and rigid transformations preserve the area of a figure. So \(A(\triangle ABC)=A(\triangle A'B'C')\)

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.