Question

Plans for a new park call for gardens directly across the sidewalk from each other to be congruent. This computer printout shows a rose garden. If the vertices of a tulip garden are located at (x1,−y1), (x2,−y2), (x3,−y3), and (x4,−y4), will the tulip garden be congruent to the rose garden?

A)No, because the transformation applied was not a rigid motion.
B)You cannot tell because you do not know where the points are on the plane.
C)Yes, because the tulip garden is a reflection of the rose garden.
D)Yes, because the same transformation was applied to all the vertices.

Answer 1

The tulip garden will be congruent to the rose garden as it is a reflection across the x-axis, preserving the size and shape of the original figure.

Step-by-step explanation:

The question involves determining whether two geometric shapes are congruent, specifically two gardens in the shape of polygons. If the vertices of the tulip garden are located at (x1, −y1), (x2, −y2), (x3, −y3), and (x4, −y4) and these represent a transformation applied to the rose garden's vertices, which we can assume to be at (x1, y1), (x2, y2), (x3, y3), and (x4, y4), then the transformation is a reflection across the x-axis. This reflection doesn't change the size or shape of the rose garden; it merely inverts the y-coordinates, thereby keeping the garden's shape and size intact but 'flipping' it symmetrically across the x-axis.

The correct answer to whether the tulip garden will be congruent to the rose garden is C: Yes, because the tulip garden is a reflection of the rose garden. Congruent figures have the same size and shape, but their orientation or position may be different. A reflection is a type of rigid motion, which preserves distances and angles, thus ensuring congruency between the two gardens.