Question

Rectangle ABCD has vertices A(-9,-6), B(-3,6), C(-3,-6), and D(-9,16). It is dilated by a scale factor of 1/3 about the origin to produce a rectangle A'B'C'D'. What is the perimeter in units of rectangle A'B'C'D'?

a) 24 units
b) 8 units
c) 16 units
d) 32 units

Answer 1

The perimeter of rectangle A'B'C'D', after dilation by a scale factor of 1/3 about the origin, is calculated to be 12 units.

Step-by-step explanation:

The question involves the concept of dilation in a coordinate plane, which is a transformation that produces an image geometrically similar to the original figure, with the distances between all points of the image and the center of dilation being a constant ratio of the distances between corresponding points of the original figure and the center of dilation. In this case, we are given that a rectangle is dilated by a scale factor of 1/3 about the origin. Thus, the coordinates of all vertices of the new rectangle A'B'C'D' are one-third of the original coordinates.

First, calculate the vertices of the new rectangle after dilation:
A'(-9/3, -6/3) = A'(-3, -2),
B'(-3/3, 6/3) = B'(-1, 2),
C'(-3/3, -6/3) = C'(-1, -2),
D'(-9/3, 16/3) = D'(-3, 16/3).

Now, we note the lengths of the sides of rectangle ABCD to find the perimeter after dilation. The length of side AB (the height of ABCD) is 6 - (-6) = 12 units, and the length of side AD (the width of ABCD) is -3 - (-9) = 6 units. Since ABCD is a rectangle, the perimeter is 2(height + width) = 2(12 + 6) = 36 units. After dilation by 1/3, the perimeter will also be reduced by a factor of 1/3: 1/3 × 36 units = 12 units.

Therefore, the perimeter of rectangle A'B'C'D' after dilation by a scale factor of 1/3 about the origin is 12 units.