Question

quadrilateral ( abcd ) is the result of translating quadrilateral ( abcd ) by 5 units to the left and 5 units down. select all of the correct statements about the unchanged properties of quadrilateral ( abcd ) and quadrilateral ( abcd ). choose all answers that apply: ( angle c ) and ( angle c ) have the same measures. the areas of quadrilateral ( abcd ) and quadrilateral ( abcd ) are the same. the perimeters of quadrilateral ( abcd ) and quadrilateral ( abcd ) are the same. none of the above

Explanation:

Step1: Properties of translation

Translation is a rigid transformation. Rigid transformations preserve side - lengths and angle - measures.

Step2: Analyze each option

• For option A: Since translation preserves angle - measures, \(\angle C\) and \(\angle C'\) have the same measures.
• For option B: Since translation preserves side - lengths, the lengths of the sides of quadrilateral \(ABCD\) and \(A'B'C'D'\) are the same. The area of a quadrilateral (which can be calculated using formulas based on side - lengths and angles, for example, if we divide it into triangles) remains the same because side - lengths and angles (used in area formulas) are preserved.
• For option C: Since translation preserves side - lengths, and the perimeter of a quadrilateral \(P = AB + BC+CD + DA\), and \(A'B'=AB\), \(B'C'=BC\), \(C'D'=CD\), \(D'A'=DA\), the perimeters of quadrilateral \(ABCD\) and \(A'B'C'D'\) are the same.

Answer:

A. \(\angle C\) and \(\angle C'\) have the same measures; B. The areas of quadrilateral \(ABCD\) and quadrilateral \(A'B'C'D'\) are the same; C. The perimeters of quadrilateral \(ABCD\) and quadrilateral \(A'B'C'D'\) are the same.