Question

answer a - c for each of the transformations below: a) identify the transformation that occurred. b) give an algebraic representation of the transformation. c) write a similarity or congruency statement for the pre - image and image.

Explanation:

Step1: Analyze transformation for 23

Observe the orientation and position change of the shape. The shape in 23 has been rotated 180 - degrees about the origin. In a 2 - D coordinate system, the algebraic rule for a 180 - degree rotation about the origin for a point (x,y) is (x,y)→( - x, - y). Since the shape and its image have the same size and shape, it is a congruency transformation. The congruency statement is pre - image KHIJ≅image K'H'I'J'.

Step2: Analyze transformation for 24

The pre - image ABCD and image A'B'C'D' in 24 show a dilation. The image is smaller than the pre - image and the center of dilation is the origin. The algebraic rule for a dilation with a scale factor k about the origin for a point (x,y) is (x,y)→(kx,ky). Here, the scale factor k is between 0 and 1. Since the shape is similar (same shape, different size), the similarity statement is pre - image ABCD∼image A'B'C'D'.

Step3: Analyze transformation for 22

In 22, the shape has been reflected across the y - axis. The algebraic rule for a reflection across the y - axis for a point (x,y) is (x,y)→( - x,y). The pre - image and image are congruent, and the congruency statement is pre - image XYZ≅image X'Y'Z'.

Answer:

22:
a) Reflection across the y - axis.
b) (x,y)→( - x,y)
c) Pre - image XYZ≅image X'Y'Z'
23:
a) 180 - degree rotation about the origin.
b) (x,y)→( - x, - y)
c) Pre - image KHIJ≅image K'H'I'J'
24:
a) Dilation about the origin.
b) (x,y)→(kx,ky) where 0 < k<1
c) Pre - image ABCD∼image A'B'C'D'