Question

geometry
name jumey
unit 4 test (b)
directions: read through and complete each of the following questions. show all work to receive full credit.
place all final answers in the spaces or boxes provided.
dilations
for each of the following, determine whether the image is a reduction or enlargement, and then determine the
scale factor.
1.

a) reduction; k = 3
b) reduction; k = 1/3
c) enlargement; k = 3
d) enlargement; k = 1/3

1. the coordinates of δabc are a(-4,5), b(2,1) and c(0, -3). triangle abc is dilated to form δabc with

coordinates a(-12, 15), b(6, 3) and c(0, -9). determine the type of dilation and the scale factor.

a. reduction; k = \\(\frac{1}{3}\\)

b. reduction; k = 3

c. enlargement; k = \\(\frac{1}{3}\\)

d. enlargement; k = 3

1. the coordinates of δabc are as follows:

a(-4, 3), b(0, -5) and c(2, 4)
a) graph the pre - image δabc.
b) dilate the pre - image by a scale factor of k = 2.
provide the new coordinates.

c) graph the image δabc.

Explanation:

Question 1

Step1: Analyze dilation type

Compare the size of the pre - image (the larger figure) and the image (the smaller figure). The image is smaller than the pre - image, so it is a reduction.

Step2: Calculate scale factor

To find the scale factor \(k\), we can look at the ratio of corresponding side lengths or the ratio of the coordinates. If we consider the horizontal or vertical distances, the pre - image has a length that is 3 times the length of the image. So the scale factor \(k=\frac{1}{3}\).

Step1: Analyze dilation type

Compare the coordinates of the pre - image \(A(-4,5),B(2,1),C(0, - 3)\) and the image \(A'(-12,15),B'(6,3),C'(0,-9)\). The absolute values of the coordinates of the image are larger than those of the pre - image, so it is an enlargement.

Step2: Calculate scale factor

For point \(A\), \(\frac{- 12}{-4}=3\), for point \(B\), \(\frac{6}{2}=3\), for point \(C\), \(\frac{-9}{-3}=3\). So the scale factor \(k = 3\).

Step1: Recall dilation rule

When a point \((x,y)\) is dilated with a scale factor \(k\) centered at the origin, the new coordinates \((x',y')\) are given by \(x'=k\times x\) and \(y'=k\times y\). Here \(k = 2\).

Step2: Calculate \(A'\)

For point \(A(-4,3)\), \(x'=2\times(-4)=-8\), \(y'=2\times3 = 6\). So \(A'(-8,6)\).

Step3: Calculate \(B'\)

For point \(B(0,-5)\), \(x'=2\times0 = 0\), \(y'=2\times(-5)=-10\). So \(B'(0,-10)\).

Step4: Calculate \(C'\)

For point \(C(2,4)\), \(x'=2\times2 = 4\), \(y'=2\times4 = 8\). So \(C'(4,8)\).

Answer:

b) reduction; \(k = \frac{1}{3}\)

Question 2