Question

scale factor and dilations
draw a line from the arrow on card a to its solution in the top corner of a card in column #2.
continue to draw lines showing the path from each card to its solution in the opposite column until
you end back at the solution on card a.

column #1
a find the scale factor used
in the dilation below.
triangle with vertices g, h, i: original sides gh=6 cm, hi=6 cm, gi=10 cm; dilated triangle g, h, i: sides gh=15 cm, hi=15 cm, gi=25 cm
48 in

c find the scale factor used
in the dilation below.
rectangle abcd: original length ab=12 cm, width bc=8 cm; dilated rectangle abcd: length ab=6 cm, width bc=4 cm
35 in

e find the scale factor used
in the dilation below.
triangle ghi: original base gi=7 cm, height gh=4 cm; dilated triangle ghi: base gi=10.5 cm, height gh=6 cm
60 in

g find the scale factor used
in the dilation below.
square r: original side=5 cm; dilated square r: side=4 cm
3 in

column #2
p the figure will be dilated by
a scale factor of 1.2. find the
new length.
rectangle: length l=40 in, width w=25 in
1/2

d the figure will be dilated by a
scale factor of 3.5. find the
new measure of the base.
triangle: base=10 in, sides=9 in
1.5

f the figure will be dilated by a
scale factor of 0.2. find the new width.
rectangle: length l=30 in, width w=?
5/2

h the figure will be dilated by a
scale factor of 5. find the new
height.
triangle: base=18 in, height=?, hypotenuse=20 in
0.8

Explanation:

To solve problems related to scale factor and dilations, we use the concept that the scale factor \( k \) is the ratio of the length of the image (\( L' \)) to the length of the original figure (\( L \)), i.e., \( k=\frac{L'}{L} \) (for finding scale factor) or \( L' = k\times L \) (for finding the new length after dilation). Let's solve a few of them:

Card A: Find the scale factor

Original triangle base \( L = 10\space\text{cm} \), Image triangle base \( L' = 25\space\text{cm} \)

Step 1: Apply scale factor formula

Scale factor \( k=\frac{L'}{L}=\frac{25}{10} \)

Step 2: Simplify the fraction

\( \frac{25}{10}=\frac{5}{2} = 2.5\) (or \( \frac{5}{2} \))

Card P: Dilate by scale factor 1.2, original length \( L = 40\space\text{in} \)

Step 1: Apply dilation formula \( L'=k\times L \)

\( L' = 1.2\times40 \)

Step 2: Calculate the product

\( 1.2\times40 = 48\space\text{in} \) (matches the solution card for A)

Card C: Find the scale factor

Original rectangle length \( L = 12\space\text{cm} \), Image rectangle length \( L' = 6\space\text{cm} \)

Step 1: Apply scale factor formula

Scale factor \( k=\frac{L'}{L}=\frac{6}{12} \)

Step 2: Simplify the fraction

\( \frac{6}{12}=\frac{1}{2}= 0.5\)

Card D: Dilate by scale factor 3.5, original base \( L = 10\space\text{in} \)

Step 1: Apply dilation formula \( L'=k\times L \)

\( L' = 3.5\times10 \)

Step 2: Calculate the product

\( 3.5\times10 = 35\space\text{in} \) (matches the solution card for C)

Card E: Find the scale factor

Original triangle base \( L = 7\space\text{cm} \), Image triangle base \( L' = 10.5\space\text{cm} \)

Step 1: Apply scale factor formula

Scale factor \( k=\frac{L'}{L}=\frac{10.5}{7} \)

Step 2: Calculate the division

\( \frac{10.5}{7}=1.5=\frac{3}{2} \)

Card F: Dilate by scale factor 0.2, original length \( L = 30\space\text{in} \) (assuming we need new width, but if original width is, say, from context, but let's check the solution. Wait, the solution card for E is 60 in? Wait, maybe miscalculation. Wait, if we consider height: original height \( 4\space\text{cm} \), image height \( 6\space\text{cm} \), scale factor \( \frac{6}{4}=1.5 \), then if we dilate a length with scale factor 1.5 to get 60 in, original length \( L=\frac{60}{1.5} = 40\space\text{in} \)? Wait, maybe better to focus on the matching.

Card G: Find the scale factor

Original square side \( L = 5\space\text{cm} \), Image square side \( L' = 4\space\text{cm} \)

Step 1: Apply scale factor formula

Scale factor \( k=\frac{L'}{L}=\frac{4}{5}=0.8 \)

Card H: Dilate by scale factor 5, original height (assuming the triangle height, say original height is, from the triangle with base 18 in, maybe height? Wait, the solution card for G is 3 in. If we dilate a length with scale factor 5 to get 15 in (wait, maybe original height is 3 in, new height \( 3\times5 = 15\space\text{in} \)? Wait, the triangle has height, maybe. But the key is using \( k=\frac{\text{image length}}{\text{original length}} \) or \( \text{new length}=k\times\text{original length} \) to match the solution cards.

Final Matches (example for Card A and P):

• Card A (find scale factor) matches Card P (dilate by 1.2, new length 48 in) because scale factor from A is \( \frac{25}{10}=2.5 \)? Wait, no, earlier calculation for P: \( 1.2\times40 = 48 \), which is the solution for A. Wait, maybe I mixed up. Let's re-express:

For Card A: We need to find scale factor. Original triangle (pre - image) has sides: \( GH = 6\space\text{cm}, HI = 6\space\text{cm}, GI = 10\space\text{cm} \). Image tr…

Step 1: Identify pre - image and image lengths

Pre - image base \( GI = 10\space\text{cm} \), Image base \( G'I' = 25\space\text{cm} \)

Step 2: Calculate scale factor

Scale factor \( k=\frac{G'I'}{GI}=\frac{25}{10}=\frac{5}{2}=2.5 \)

Step 1: Identify scale factor and original length

Scale factor \( k = 1.2 \), Original length \( L = 40\space\text{in} \)

Step 2: Calculate new length

New length \( L'=k\times L=1.2\times40 = 48\space\text{in} \)

Answer:

The scale factor is \( \boldsymbol{\frac{5}{2}} \) (or \( 2.5 \))

If we take Card P: