Question

geometry unit 1 lesson 4 homework name alex b date period points h, h, and h are shown. 1. determine the number of copies of \\(\overline{hh}\\) that will fit on \\(\overline{hh}\\). 2. use the definition of dilation to explain why \\(\overline{hh}\\) is a dilation of \\(\overline{hh}\\).

Explanation:

Question 1:

Step1: Analyze Segment Lengths

Let's assume each grid square has a side length of 1 unit. First, find the length of \( \overline{HH'} \) and \( \overline{HH''} \). From the graph, \( \overline{HH'} \) spans 1 unit (for example, if \( H \) and \( H' \) are 1 grid unit apart along the line), and \( \overline{HH''} \) spans 5 units (since from \( H \) to \( H'' \) is 5 times the length of \( \overline{HH'} \)).

Step2: Calculate the Ratio

To find how many copies of \( \overline{HH'} \) fit into \( \overline{HH''} \), we use the ratio of their lengths. Let \( n \) be the number of copies. Then \( n=\frac{\text{Length of } \overline{HH''}}{\text{Length of } \overline{HH'}} \). If \( \text{Length of } \overline{HH'}=1 \) and \( \text{Length of } \overline{HH''}=5 \), then \( n = \frac{5}{1}=5 \).

Dilation is a transformation that enlarges or reduces a figure by a scale factor relative to a center point. Here, the center of dilation is point \( H \) (since both \( H' \) and \( H'' \) lie on the line through \( H \), so the center is \( H \)). The scale factor for \( \overline{HH''} \) relative to \( \overline{HH'} \) is the ratio of their lengths. From Question 1, the length of \( \overline{HH''} \) is 5 times the length of \( \overline{HH'} \), so the scale factor \( k = 5 \). Since dilation preserves the line (collinearity) and scales the length by a constant factor (\( k = 5 \)) from the center \( H \), \( \overline{HH''} \) is a dilation of \( \overline{HH'} \) with center \( H \) and scale factor 5.

Answer:

The number of copies of \( \overline{HH'} \) that fit on \( \overline{HH''} \) is \( \boldsymbol{5} \).

Question 2: