Finding Scale Factor Of Similar Figures

🔑 Key Concepts

• A scale factor compares a measurement in the new figure with its corresponding measurement in the original figure.
• The scale-factor rule is \(\displaystyle k=\frac{\text{new}}{\text{original}}\).
• The direction of the scaling matters. Scaling figure \(A\) to figure \(B\) may produce a different scale factor than scaling \(B\) to \(A\).
• Scale factors in opposite directions are reciprocals.
• A scale factor greater than \(1\) produces an enlargement.
• A scale factor between \(0\) and \(1\) produces a reduction.
• All perfect circles are similar, so their diameters or radii may be compared to determine the scale factor.
• Once a scale factor is known, it can be multiplied by other corresponding side lengths to find missing measurements.

✏️ Worked Examples

Finding Scale Factors in Either Direction

For every problem, first identify which figure is the original and which figure is the new image. Then compare corresponding measurements in the correct order:

\(\displaystyle k=\frac{\text{new measurement}}{\text{original measurement}}\)

When the direction is reversed, the numerator and denominator switch places. Therefore, the two scale factors are reciprocals.

Example 1: Triangles \(A\) and \(B\)

Triangle \(A\) has a corresponding base length of \(8\) units, while triangle \(B\) has a base length of \(4\) units.

Scale triangle \(A\) to triangle \(B\):

\(\displaystyle k=\frac{B}{A}\)

\(\displaystyle k=\frac{4}{8}\)

\(\displaystyle k=\frac12=0.5\)

\(\boxed{k=\frac12}\)

Triangle \(B\) is one-half the linear size of triangle \(A\), so this is a reduction.

Reverse the direction: \(B\) to \(A\)

\(\displaystyle k=\frac{A}{B}\)

\(\displaystyle k=\frac{8}{4}=2\)

\(\boxed{k=2}\)

The scale factors \(\frac12\) and \(2\) are reciprocals.

Example 2: Circles \(A\) and \(B\)

Every perfect circle is similar to every other perfect circle. Therefore, we may compare their corresponding diameters or their corresponding radii.

Circle \(A\) has a diameter of \(8\) centimeters, and circle \(B\) has a diameter of \(12\) centimeters.

Scale circle \(A\) to circle \(B\):

\(\displaystyle k=\frac{12}{8}\)

\(\displaystyle k=\frac32=1.5\)

\(\boxed{k=\frac32}\)

Circle \(B\) is \(1.5\) times the linear size of circle \(A\), or \(150\%\) of its size.

Reverse the direction: \(B\) to \(A\)

\(\displaystyle k=\frac{8}{12}\)

\(\displaystyle k=\frac23\)

\(\boxed{k=\frac23}\)

The scale factors \(\frac32\) and \(\frac23\) are reciprocals.

Example 3: Find \(\overline{AB}\) and \(\overline{EF}\)

The similarity statement \(\triangle ABC\sim\triangle DEF\) identifies the correspondence:

\(A\leftrightarrow D,\quad B\leftrightarrow E,\quad C\leftrightarrow F\)

The corresponding vertical sides measure \(12\) centimeters and \(6\) centimeters.

\(\displaystyle k_{\text{large}\to\text{small}}=\frac{6}{12}=\frac12\)

\(\displaystyle k_{\text{small}\to\text{large}}=\frac{12}{6}=2\)

Find \(\overline{AB}\)

The smaller base \(\overline{DE}\) is \(2.5\) centimeters. Scale from the smaller triangle to the larger triangle using \(k=2\).

\(AB=2(DE)\)

\(AB=2(2.5)\)

\(\boxed{AB=5\text{ cm}}\)

Find \(\overline{EF}\)

The larger hypotenuse \(\overline{BC}\) is \(13\) centimeters. Scale from the larger triangle to the smaller triangle using \(k=\frac12\).

\(\displaystyle EF=\frac12(BC)\)

\(\displaystyle EF=\frac12(13)\)

\(\boxed{EF=6.5\text{ cm}}\)

Why the Two Directions Produce Reciprocal Scale Factors

Triangle Example

\(\displaystyle A\to B:\frac48=\frac12\)

\(\displaystyle B\to A:\frac84=2\)

\(\displaystyle \frac12\cdot2=1\)

Circle Example

\(\displaystyle A\to B:\frac{12}{8}=\frac32\)

\(\displaystyle B\to A:\frac8{12}=\frac23\)

\(\displaystyle \frac32\cdot\frac23=1\)

General Rule

\(\displaystyle k_{A\to B}=\frac{B}{A}\)

\(\displaystyle k_{B\to A}=\frac{A}{B}\)

\(\displaystyle k_{A\to B}\cdot k_{B\to A}=1\)

Reversing the direction of a scaling switches the old and new measurements. This reverses the fraction, producing its reciprocal.

Use the Same Scale Factor for Every Corresponding Length

In the final problem, the known pair \(12\) centimeters and \(6\) centimeters determines the scale factor. That same factor must relate every other pair of corresponding sides.

Small Triangle to Large Triangle

\(k=2\)

\(DE\leftrightarrow AB\)

\(AB=2(2.5)=5\text{ cm}\)

Large Triangle to Small Triangle

\(\displaystyle k=\frac12\)

\(BC\leftrightarrow EF\)

\(\displaystyle EF=\frac12(13)=6.5\text{ cm}\)

🧠 Math Vocabulary

• Scale factor: The multiplier that changes every length of an original figure to create a scaled copy.
• Original figure: The figure being scaled.
• Image: The new figure produced by a transformation.
• Scaled copy: A figure created by multiplying all lengths of an original figure by the same scale factor.
• Similar figures: Figures with congruent corresponding angles and proportional corresponding side lengths.
• Corresponding sides: Sides that occupy matching positions in similar figures.
• Corresponding vertices: Vertices that occupy matching positions in similar figures.
• Ratio: A comparison of two quantities using division.
• Proportion: An equation showing that two ratios are equivalent.
• Reciprocal: A number formed by switching the numerator and denominator of a nonzero fraction.
• Enlargement: A scaled copy produced by a scale factor greater than \(1\).
• Reduction: A scaled copy produced by a scale factor between \(0\) and \(1\).
• Diameter: A segment passing through the center of a circle with endpoints on the circle.
• Radius: A segment from the center of a circle to a point on the circle.
• Hypotenuse: The side opposite the right angle in a right triangle.
• Similarity statement: A statement that lists corresponding vertices of similar figures in matching order.

💡 Main Idea

The scale factor depends on the direction of the transformation. To scale from figure \(A\) to figure \(B\), divide a measurement in \(B\) by its corresponding measurement in \(A\). Reversing the direction reverses the ratio, so the two scale factors are reciprocals. Once the scale factor is known, it may be applied to every pair of corresponding side lengths.

📚 What You Should Already Know

Students should know how to identify corresponding measurements, write and simplify ratios, multiply fractions and decimals, recognize reciprocal fractions, and interpret the order of vertices in a similarity statement.

🚀 What Comes Next

Students can extend this reasoning by finding missing side lengths in more complex similar polygons, writing and solving proportions, drawing scaled copies on a coordinate plane, and investigating how scale factor affects perimeter and area.

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