Rectangle \(A\) measures \(20\) centimeters by \(5\) centimeters.
The ratio of length to width is \(\frac{20}{5}=4\).
Every scaled copy must preserve this same length-to-width ratio.
The dimensions of a scaled copy may be larger or smaller than the original dimensions.
The same scale factor must multiply both the length and width.
Possible dimensions include \(6\) by \(1.5\), \(18\) by \(4.5\), and \(80\) by \(20\).
Dimensions \(10\) by \(2\) and \(11\) by \(4\) do not preserve the original ratio.
✏️ Worked Example
Problem
Rectangle \(A\) measures \(20\) centimeters by \(5\) centimeters. Rectangle \(B\) is a scaled copy of Rectangle \(A\). Select all measurement pairs that could be the dimensions of Rectangle \(B\).
A. \(6\text{ cm}\) by \(1.5\text{ cm}\)
B. \(10\text{ cm}\) by \(2\text{ cm}\)
C. \(11\text{ cm}\) by \(4\text{ cm}\)
D. \(18\text{ cm}\) by \(4.5\text{ cm}\)
E. \(80\text{ cm}\) by \(20\text{ cm}\)
Original Rectangle
The original length-to-width ratio is:
\(\displaystyle \frac{20}{5}=4\)
Therefore, each possible scaled copy must also have a length-to-width ratio of \(4\).
Choice A: \(6\) by \(1.5\)
Compare the length to the width:
\(\displaystyle \frac{6}{1.5}=4\)
This matches the original ratio:
\(\displaystyle \frac{20}{5}=\frac{6}{1.5}=4\)
The scale factor from Rectangle \(A\) to this rectangle is:
\(\displaystyle k=\frac{6}{20}=0.3\)
\(\displaystyle \frac{1.5}{5}=0.3\)
\(\boxed{\text{Yes, this is a scaled copy.}}\)
Choice B: \(10\) by \(2\)
Compare the length to the width:
\(\displaystyle \frac{10}{2}=5\)
The original ratio is \(4\), not \(5\):
\(\displaystyle \frac{20}{5}\ne\frac{10}{2}\)
The separate scale factors also disagree:
\(\displaystyle \frac{10}{20}=\frac{1}{2}\)
\(\displaystyle \frac{2}{5}=\frac{2}{5}\)
\(\displaystyle \frac{1}{2}\ne\frac{2}{5}\)
\(\boxed{\text{No, this is not a scaled copy.}}\)
Choice C: \(11\) by \(4\)
Compare the length to the width:
\(\displaystyle \frac{11}{4}=2.75\)
This does not match the original ratio:
\(\displaystyle 2.75\ne4\)
The scale factors are also different:
\(\displaystyle \frac{11}{20}=0.55\)
\(\displaystyle \frac{4}{5}=0.8\)
\(0.55\ne0.8\)
\(\boxed{\text{No, this is not a scaled copy.}}\)
Choice D: \(18\) by \(4.5\)
Compare the length to the width:
\(\displaystyle \frac{18}{4.5}=4\)
This matches the original ratio:
\(\displaystyle \frac{20}{5}=\frac{18}{4.5}=4\)
The same scale factor changes both dimensions:
\(\displaystyle \frac{18}{20}=0.9\)
\(\displaystyle \frac{4.5}{5}=0.9\)
\(\boxed{\text{Yes, this is a scaled copy.}}\)
Choice E: \(80\) by \(20\)
Compare the length to the width:
\(\displaystyle \frac{80}{20}=4\)
This matches the original ratio:
\(\displaystyle \frac{20}{5}=\frac{80}{20}=4\)
Both original dimensions were multiplied by \(4\):
\(20(4)=80\)
\(5(4)=20\)
\(\boxed{\text{Yes, this is a scaled copy.}}\)
Final Answer
The measurement pairs that preserve the original length-to-width ratio of \(4\) are:
\(\boxed{6\text{ cm by }1.5\text{ cm}}\)
\(\boxed{18\text{ cm by }4.5\text{ cm}}\)
\(\boxed{80\text{ cm by }20\text{ cm}}\)
\(\boxed{\text{A, D, and E}}\)
Three Ways to Check a Possible Scaled Copy
Method 1: Compare Length to Width
Calculate the length-to-width ratio for both rectangles.
If both ratios produce the same multiplier, the new rectangle is a scaled copy.
Method 3: Use Cross Products
A rectangle with dimensions \(L\) and \(W\) is a scaled copy when:
\(\displaystyle \frac{20}{5}=\frac{L}{W}\)
Cross multiply:
\(20W=5L\)
Divide by \(5\):
\(\boxed{L=4W}\)
The length must be exactly four times the width.
🧠 Math Vocabulary
Scaled copy: A figure created by multiplying every length of an original figure by the same scale factor.
Scale factor: The constant multiplier used to enlarge or reduce a figure.
Similar figures: Figures with the same shape, proportional corresponding side lengths, and congruent corresponding angles.
Corresponding sides: Sides that occupy matching positions in two figures.
Ratio: A comparison of two quantities using division.
Equivalent ratios: Ratios that represent the same comparison.
Proportion: An equation stating that two ratios are equivalent.
Proportional: Having corresponding measurements related by a constant ratio.
Constant of proportionality: The constant multiplier relating corresponding quantities.
Reduction: A scaled copy produced by a scale factor between \(0\) and \(1\).
Enlargement: A scaled copy produced by a scale factor greater than \(1\).
Dimensions: Measurements describing the size of a figure, such as length and width.
Cross products: Products formed by multiplying diagonally across a proportion.
Length-to-width ratio: A comparison of a rectangle’s length to its width.
💡 Main Idea
Rectangle \(A\) has a length-to-width ratio of \(\frac{20}{5}=4\). Any scaled copy must preserve this ratio, meaning its length must also be four times its width. The dimensions \(6\) by \(1.5\), \(18\) by \(4.5\), and \(80\) by \(20\) all satisfy this condition.
📚 What You Should Already Know
Students should know how to write and simplify ratios, divide whole numbers and decimals, identify corresponding dimensions, solve proportions, and determine whether two ratios are equivalent.
🚀 What Comes Next
Students can extend this reasoning by finding missing dimensions in scaled figures, determining scale factors, drawing scaled copies, testing polygons for similarity, and studying how scaling affects perimeter and area.
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