Question

geometry
unit 1
lesson 4 homework
name __________________
date ___________________
period ________________

points ( h ), ( h ), and ( h ) are shown. image of a grid with points ( h ), ( h ), ( h ) connected by a line

1. determine the number of copies of ( overline{hh} ) that will fit on ( overline{hh} ).

1. use the definition of dilation to explain why ( overline{hh} ) is a dilation of ( overline{hh} ).

1. complete the statement.

point ( \begin{cases} h \\ h \\ h end{cases} ) partitions ( \begin{cases} overline{hh} \\ overline{hh} \\ overline{hh} end{cases} ) so that the ratio of ( \frac{hh}{hh} ) is ( \begin{cases} \frac{1}{4}, \frac{1}{5} end{cases} ) options in boxes

use the pythagorean theorem to complete the table.

4.
5.
6.

segment	length
( overline{hh} )
( overline{hh} )

Explanation:

Question 1: Determine the number of copies of $\boldsymbol{\overline{HH'}}$ that will fit on $\boldsymbol{\overline{HH''}}$.

Step1: Analyze the grid

Assume each grid unit is 1. Visually, count the number of times the length of $\overline{HH'}$ fits into $\overline{HH''}$.

Step2: Count the segments

If $\overline{HH'}$ is one "unit" segment, and $\overline{HH''}$ is three times as long (by counting grid intervals or visual proportion), then 3 copies of $\overline{HH'}$ fit into $\overline{HH''}$.

Question 2: Use the definition of dilation to explain why $\boldsymbol{\overline{HH''}}$ is a dilation of $\boldsymbol{\overline{HH'}}$.

Dilation requires a center (here, $H$), a scale factor, and collinear points (same line). $\overline{HH''}$ and $\overline{HH'}$ share endpoint $H$, lie on the same line, and $HH''$ is a multiple (scale factor > 1) of $HH'$, satisfying dilation’s definition (enlargement about $H$).

Question 3: Complete the statement.

Point $\boldsymbol{H'}$ partitions $\boldsymbol{\overline{HH''}}$ so that the ratio of $\boldsymbol{\frac{HH'}{HH''}}$ is $\boldsymbol{\frac{1}{3}}$.

From Question 1, $HH'' = 3 \cdot HH'$, so $\frac{HH'}{HH''} = \frac{HH'}{3 \cdot HH'} = \frac{1}{3}$. The point $H'$ lies on $\overline{HH''}$, so it partitions the segment, and the ratio of the shorter part ($HH'$) to the whole ($HH''$) is $\frac{1}{3}$.

Questions 4–6: Use the Pythagorean theorem to complete the table.

Assume the grid has horizontal/vertical spacing of 1 unit. Let’s define coordinates:

• Let $H = (0, 0)$, $H' = (4, 1)$ (approximate from grid), $H'' = (12, 3)$ (since $HH''$ is 3× $HH'$).

Question 4: Length of $\boldsymbol{\overline{HH'}}$

Answer:

3