Question

suppose translating quadrilateral abcd by 8 units to the left and 8 units down results in a new quadrilateral abcd. which statements are correct? a a and a have the same coordinates. b segment cd is parallel to segment cd. c ∠b and ∠b have the different measures. d the areas of quadrilateral abcd and quadrilateral abcd are the same. e the perimeters of quadrilateral abcd and quadrilateral abcd are the same

Explanation:

Step1: Analyze translation properties

Translation is a rigid transformation. Rigid transformations preserve shape, size, angle measures, parallelism, and perpendicularity.

Step2: Check each option

• Option A:

When translating a point \(A\) by \(8\) units left and \(8\) units down, its coordinates change. If \(A=(x,y)\), then \(A'=(x - 8,y-8)\). So \(A\) and \(A'\) do not have the same coordinates.

• Option B:

Since translation preserves parallelism, if \(CD\) is a segment in the original quadrilateral and \(C'D'\) is the corresponding segment in the translated quadrilateral, then \(CD\parallel C'D'\).

• Option C:

Translation preserves angle measures. So \(m\angle B=m\angle B'\).

• Option D:

Since translation is a rigid transformation and preserves the size of the figure, the area of \(ABCD\) is equal to the area of \(A'B'C'D'\).

• Option E:

Since translation preserves the lengths of the sides (because it is a rigid transformation), the perimeter of \(ABCD\) (sum of side - lengths \(AB + BC+CD + DA\)) is equal to the perimeter of \(A'B'C'D'\) (sum of side - lengths \(A'B'+B'C'+C'D'+D'A'\)) as \(AB = A'B'\), \(BC = B'C'\), \(CD = C'D'\), \(DA=D'A'\)

Answer:

B. Segment \(CD\) is parallel to segment \(C'D'\); D. The areas of quadrilateral \(ABCD\) and quadrilateral \(A'B'C'D'\) are the same; E. The perimeters of quadrilateral \(ABCD\) and quadrilateral \(A'B'C'D'\) are the same.