Question

geometry unit 1 lesson 4 homework name 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}\\). 3. complete the statement. sections with point, partitions, ratio boxes

Explanation:

Problem 1: Determine the number of copies of \( \overline{HH'} \) that will fit on \( \overline{HH''} \)

Step 1: Analyze the segment lengths

From the graph, we can observe the relative lengths of \( \overline{HH'} \) and \( \overline{HH''} \). Let's assume the grid has equal spacing. By counting the number of units (or the ratio of lengths), we see that \( \overline{HH''} \) is 4 times the length of \( \overline{HH'} \) (since \( H' \) divides \( \overline{HH''} \) into 4 equal parts? Wait, actually, looking at the graph, if we consider the distance from \( H \) to \( H' \) and \( H \) to \( H'' \), let's check the ratio. Suppose the length of \( \overline{HH'} \) is \( l \), then \( \overline{HH''} \) is \( 4l \)? Wait, no, maybe 4? Wait, let's see: from \( H \) to \( H' \) is 1 part, and from \( H \) to \( H'' \) is 4 parts? Wait, maybe the number of copies is 4? Wait, no, let's re-examine. Wait, the segment \( \overline{HH''} \) has \( H' \) as a point such that \( HH' \) is a fraction of \( HH'' \). Let's count the number of times \( HH' \) fits into \( HH'' \). If we look at the grid, the distance from \( H \) to \( H' \) is 1 unit (in terms of the grid's segment), and from \( H \) to \( H'' \) is 4 units? Wait, no, maybe the ratio is 4? Wait, actually, in the graph, the line from \( H \) to \( H'' \) passes through \( H' \), and the number of segments \( HH' \) in \( HH'' \) is 4? Wait, no, let's see: if \( H' \) is 1/4 of the way? No, maybe the length of \( HH' \) is 1, and \( HH'' \) is 4, so 4 copies? Wait, maybe I made a mistake. Wait, let's think again. The problem is about dilation, so the ratio of \( HH' \) to \( HH'' \) is 1:4? No, wait, the number of copies of \( HH' \) that fit into \( HH'' \) is the ratio of \( HH'' \) length to \( HH' \) length. From the graph, if \( HH' \) is length \( x \), then \( HH'' \) is length \( 4x \), so 4 copies? Wait, no, maybe 3? Wait, the user's graph: let's assume that from \( H \) to \( H' \) is 1 unit, and from \( H \) to \( H'' \) is 4 units? Wait, maybe the correct answer is 4? Wait, no, let's check the grid. Suppose each square is 1 unit. Let's find the coordinates. Let's assume \( H \) is at (0, 4), \( H' \) is at (1, 3), and \( H'' \) is at (4, 0). Then the distance \( HH' \) is \( \sqrt{(1-0)^2 + (3-4)^2} = \sqrt{1 + 1} = \sqrt{2} \). The distance \( HH'' \) is \( \sqrt{(4-0)^2 + (0-4)^2} = \sqrt{16 + 16} = \sqrt{32} = 4\sqrt{2} \). So the ratio \( HH'' / HH' = 4\sqrt{2} / \sqrt{2} = 4 \). So 4 copies of \( HH' \) fit into \( HH'' \).

Step 2: Confirm the ratio

Since the length of \( \overline{HH''} \) is 4 times the length of \( \overline{HH'} \), the number of copies of \( \overline{HH'} \) that fit on \( \overline{HH''} \) is 4.

Dilation is a transformation that changes the size of a figure but not its shape, by a scale factor relative to a center (here, the center is \( H \), since both segments start at \( H \)). The scale factor \( k \) is the ratio of the length of the image segment (\( \overline{HH''} \)) to the length of the original segment (\( \overline{HH'} \)). From Problem 1, we found the scale factor \( k = \frac{HH''}{HH'} = 4 \). Since dilation multiplies the length of the original segment by the scale factor (and preserves the direction, as both segments are colinear from \( H \) through \( H' \) to \( H'' \)), \( \overline{HH''} \) is a dilation of \( \overline{HH'} \) with center \( H \) and scale factor 4.

From the graph, \( H' \) is a point on \( \overline{HH''} \) that divides it such that \( \overline{HH'} \) is part of \( \overline{HH''} \). The ratio \( \frac{HH'}{HH''} \) is \( \frac{1}{4} \) (since \( HH'' = 4 \cdot HH' \)). So the point \( H' \) partitions \( \overline{HH''} \) with the ratio \( \frac{HH'}{HH''} = \frac{1}{4} \).

Answer:

4

Problem 2: Use the definition of dilation to explain why \( \overline{HH''} \) is a dilation of \( \overline{HH'} \)