SAT — mathHard3 min read

Similar Triangles and Proportional Reasoning

Use similar triangles to find missing sides on the Digital SAT. Apply AA similarity, set up proportions, and work with scale factors.

Tutor AI Team2026-02-25T01:33:50.839Z

Similar triangles have the same shape but different sizes. Their corresponding angles are equal, and their corresponding sides are proportional. The Digital SAT uses similar triangles extensively — in standalone problems, in figures with parallel lines creating similar triangles, and in real-world applications like shadow problems.

Core Concepts

What Makes Triangles Similar?

AA (Angle-Angle) Similarity: If two angles of one triangle equal two angles of another, the triangles are similar.

Since angle sums are always 180°, matching two angles automatically matches all three.

Corresponding Sides and Scale Factor

If $\triangle ABC \sim \triangle DEF$ :

$\frac{AB}{DE} = \frac{BC}{EF} = \frac{AC}{DF} = k$

where $k$ is the scale factor.

Setting Up Proportions

Identify corresponding sides (match by opposite equal angles).
Set up the proportion.
Cross-multiply and solve.

Area and Volume with Scale Factor

Area ratio = $k^2$ (scale factor squared)
Volume ratio = $k^3$ (scale factor cubed)

If the scale factor is 3, the area ratio is 9 and the volume ratio is 27.

Common Configurations

Parallel line in a triangle: creates two similar triangles.
Shadow problems: person/object and their shadow form similar triangles with the sun's rays.
Nested triangles: a line parallel to one side cuts the other two sides proportionally.

Strategy Tips

Tip 1: Match Angles First

Identify which angles correspond, then match the opposite sides.

Tip 2: Write the Similarity Statement Carefully

$\triangle ABC \sim \triangle DEF$ means $A ↔ D$ , $B ↔ E$ , $C ↔ F$ . Corresponding vertices must be in order.

Tip 3: Check Your Proportion Direction

Make sure smaller-triangle sides are consistently on one side of the proportion.

Worked Example: Example 1

Problem

$\triangle ABC \sim \triangle DEF$ . $AB = 6$ , $DE = 9$ , $BC = 8$ . Find $EF$ .

Scale factor: $\frac{9}{6} = 1.5$ .

$EF = 8 \times 1.5 = 12$

Solution

Worked Example: Example 2

Problem

A tree casts a shadow 15 ft long. A 6-ft person nearby casts a shadow 4 ft long. How tall is the tree?

Similar triangles: $\frac{h}{15} = \frac{6}{4}$ → $h = \frac{90}{4} = 22.5$ ft.

Solution

Worked Example: SAT-Style

Problem

In triangle $ABC$ , $DE$ is parallel to $BC$ where $D$ is on $AB$ and $E$ is on $AC$ . $AD = 4$ , $DB = 6$ , $DE = 5$ . Find $BC$ .

$\frac{AD}{AB} = \frac{DE}{BC}$ . $AB = 10$ .

$\frac{4}{10} = \frac{5}{BC}$ → $BC = 12.5$

Solution

Worked Example: Example 4

Problem

Two similar triangles have a scale factor of 2:5. If the smaller has area 12, find the larger's area.

Area ratio: $(5/2)^2 = 6.25$ . Larger area: $12 \times 6.25 = 75$ .

Solution

Practice Problems

Problem 1

$\triangle PQR \sim \triangle XYZ$ . $PQ = 10$ , $XY = 15$ , $QR = 12$ . Find $YZ$ .

Problem 2

A 5-ft stick casts a 3-ft shadow. A building casts a 24-ft shadow. How tall is the building?

Problem 3

Similar triangles with scale factor 3:1. Small triangle has perimeter 18. Find large triangle's perimeter.

Problem 4

Scale factor is 4. Area of small triangle is 10. Area of large triangle?

Want to check your answers and get step-by-step solutions?

Common Mistakes

Mismatching corresponding sides. Always verify which sides correspond by checking opposite angles.
Using the wrong scale factor direction. If small:large = 2:5, the scale factor FROM small TO large is 5/2.
Forgetting to square for area. If sides scale by $k$ , areas scale by $k^2$ .
Assuming triangles are similar without proof. Check for AA similarity.

Key Takeaways

✓
AA similarity: two equal angles → similar triangles.
✓
Corresponding sides are proportional: set up cross-multiplication.
✓
Scale factor applies to all corresponding lengths.
✓
Area scales by $k^2$ ; volume by $k^3$ .
✓
Shadow problems and parallel lines are common similar-triangle setups.

Tags:

#sat#math#geometry#similar-triangles#proportions

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Core Concepts

What Makes Triangles Similar?

Corresponding Sides and Scale Factor

Setting Up Proportions

Area and Volume with Scale Factor

Common Configurations

Strategy Tips

Tip 1: Match Angles First

Tip 2: Write the Similarity Statement Carefully

Tip 3: Check Your Proportion Direction

Worked Example: Example 1

Worked Example: Example 2

Worked Example: SAT-Style

Worked Example: Example 4

Practice Problems

Problem 1

Problem 2

Problem 3

Problem 4

Common Mistakes

Key Takeaways

Related Topics

Parallel Structure

Verb Tense and Consistency

Modifier Placement and Dangling Modifiers

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