Identify Similar Shapes

📝 Practice Worksheets

Are the Shapes Proportional? Drawing Scale Figures Similar Polygons: Missing Side Length Finding the Scale Factor

🛠️ Related Tools

Scale Factor Practice Lab Similarity Explorer

🔑 Key Concepts

Similar figures have the same shape, although they may have different sizes.
Corresponding angles in similar figures are congruent.
Corresponding side lengths in similar figures are proportional.
Equivalent ratios produce the same scale factor.
To test for similarity, compare corresponding sides in the same order.
If every pair of corresponding side lengths has the same ratio, the figures are similar.
If even one pair produces a different ratio, the figures are not similar.

✏️ Worked Examples

Example 1: Similar Right Triangles

Determine whether the two right triangles are similar.

Match the corresponding vertical sides and horizontal sides:

\(\displaystyle \frac{12}{9} \quad\text{and}\quad \frac{8}{6} \)

Simplify both ratios:

\(\displaystyle \frac{12}{9}=\frac{4}{3} \)

\(\displaystyle \frac{8}{6}=\frac{4}{3} \)

The ratios of corresponding sides are equivalent:

\(\displaystyle \frac{12}{9}=\frac{8}{6} \)

\(\boxed{\text{The triangles are similar.}}\)

The scale factor from the smaller triangle to the larger triangle is:

\(\displaystyle k=\frac{4}{3}\)

Example 2: Rectangles That Are Not Similar

Determine whether the two rectangles are similar.

Compare the corresponding widths and heights:

\(\displaystyle \frac{12}{10} \quad\text{and}\quad \frac{8}{7} \)

Simplify the first ratio:

\(\displaystyle \frac{12}{10}=\frac{6}{5} \)

Compare the two ratios:

\(\displaystyle \frac{6}{5}\ne\frac{8}{7} \)

Cross products also confirm that the ratios are not equivalent:

\(12(7)=84\)

\(10(8)=80\)

\(84\ne80\)

\(\boxed{\text{The rectangles are not similar.}}\)

Because the corresponding sides do not share one constant scale factor, one rectangle is not a scaled copy of the other.

A Reliable Process for Testing Similarity

1. Match Corresponding Sides

Identify sides that occupy the same relative position in each figure.

2. Write Ratios Consistently

Compare larger to smaller for every pair, or smaller to larger for every pair.

3. Check for One Scale Factor

If all corresponding ratios are equivalent, the figures are similar.

Using Cross Products to Test Equivalent Ratios

For two ratios \(\frac{a}{b}\) and \(\frac{c}{d}\), the ratios are equivalent when their cross products are equal:

\(\displaystyle \frac{a}{b}=\frac{c}{d} \quad\Longleftrightarrow\quad ad=bc \)

For the triangles:

\(\displaystyle \frac{12}{9}=\frac{8}{6} \)

\(12(6)=72\)

\(9(8)=72\)

\(\boxed{72=72}\)

For the rectangles:

\(\displaystyle \frac{12}{10}\ne\frac{8}{7} \)

\(12(7)=84\)

\(10(8)=80\)

\(\boxed{84\ne80}\)

Keep the Order of the Ratios Consistent

A correct comparison keeps the same figure in the numerator of every ratio and the other figure in the denominator:

\(\displaystyle \frac{\text{large vertical side}}{\text{small vertical side}} = \frac{\text{large horizontal side}}{\text{small horizontal side}} \)

\(\displaystyle \frac{12}{9} = \frac{8}{6} \)

Reversing only one ratio would compare the sides inconsistently and could produce an incorrect conclusion.

🧠 Math Vocabulary

Similar figures: Figures with the same shape, proportional corresponding side lengths, and congruent corresponding angles.
Similarity: The relationship between figures that have the same shape but not necessarily the same size.
Corresponding: Occupying the same relative position in two figures.
Corresponding sides: Sides that match in position between two figures.
Corresponding angles: Angles that match in position between 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 quantities related by one constant ratio.
Scale factor: The constant multiplier used to enlarge or reduce a figure.
Cross products: The products formed by multiplying diagonally across a proportion.
Congruent angles: Angles with equal measures.
Scaled copy: A figure produced by multiplying every length of an original figure by the same scale factor.
Reduction: A scaled copy with a scale factor between \(0\) and \(1\).
Enlargement: A scaled copy with a scale factor greater than \(1\).

💡 Main Idea

To determine whether two figures are similar, match their corresponding sides and compare the side lengths using ratios. The figures are similar only when every pair of corresponding side lengths produces the same scale factor. The triangles with side pairs \(12\) and \(9\), and \(8\) and \(6\), are similar because both ratios simplify to \(\frac{4}{3}\). The \(12\)-by-\(8\) and \(10\)-by-\(7\) rectangles are not similar because their corresponding ratios are not equivalent.

📚 What You Should Already Know

Students should know how to write and simplify ratios, identify corresponding parts of figures, solve proportions, compare fractions, and use cross products to determine whether two ratios are equivalent.

🚀 What Comes Next

Students can extend this reasoning by finding missing side lengths in similar figures, determining scale factors, drawing scaled copies, comparing perimeter and area under dilation, and using similarity to solve indirect-measurement problems.

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