Question

unit 4 test (a)
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 dilation is a reduction or enlargement, and then determine the
scale factor.
1.
a) reduction; k = ½
b) reduction; k = 2
c) enlargement; k = ½
d) enlargement; k = 2

1. the coordinates of δabc are a(-3,24), b(6,12) and c(0,-9). triangle abc is dilated to form δabc

with coordinates a(-1,8), b(2,4) and c(0,-3). determine the type of dilation and the scale factor.
a. enlargement; k = \\(\frac{1}{3}\\) c. reduction; k = \\(\frac{1}{3}\\)
b. enlargement; k = 3 d. reduction; k = 3

1. the coordinates of δabc are as follows:

a(-4,2), b(1,3) and c(0,-5)
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 Triangle Sizes

Compare the original triangle (FGH) and the dilated triangle (F'G'H'). The original triangle has a base (from F to H) of length 2 units (grid squares), and the dilated one has a base of 1 unit. So it's a reduction.

Step2: Calculate Scale Factor

Scale factor \( k = \frac{\text{length of image}}{\text{length of pre - image}}=\frac{1}{2} \). So the dilation is a reduction with \( k=\frac{1}{2} \).

Step1: Check Coordinate Ratios

For point A: \( \frac{- 1}{-3}=\frac{1}{3} \), \( \frac{8}{24}=\frac{1}{3} \). For point B: \( \frac{2}{6}=\frac{1}{3} \), \( \frac{4}{12}=\frac{1}{3} \). For point C: \( \frac{0}{0} \) (undefined, but other coordinates show ratio), \( \frac{-3}{-9}=\frac{1}{3} \).

Step2: Determine Dilation Type

Since the scale factor \( k = \frac{1}{3}<1 \), it's a reduction.

Step1: Recall Dilation Formula

For a dilation with scale factor \( k \), the new coordinates \( (x',y')=k(x,y) \), where \( (x,y) \) are original coordinates. Here \( k = 2 \).

Step2: Calculate \( A' \)

For \( A(-4,2) \), \( A'=(2\times(-4),2\times2)=(-8,4) \)

Step3: Calculate \( B' \)

For \( B(1,3) \), \( B'=(2\times1,2\times3)=(2,6) \)

Step4: Calculate \( C' \)

For \( C(0, - 5) \), \( C'=(2\times0,2\times(-5))=(0,-10) \)

Answer:

a) reduction; \( k=\frac{1}{2} \)

Question 2