Proportional Geometry Examples in Math

Start with the recap, study the fully worked examples, then use the practice problems to check your understanding of Proportional Geometry.

This page combines explanation, solved examples, and follow-up practice so you can move from recognition to confident problem-solving in Math.

Concept Recap

Using ratios and proportions to relate corresponding measurements in similar or scaled geometric figures.

Similar triangles have proportional sides: if one side doubles, all sides double.

Read the full concept explanation →

How to Use These Examples

  • Read the first worked example with the solution open so the structure is clear.
  • Try the practice problems before revealing each solution.
  • Use the related concepts and background knowledge badges if you feel stuck.

What to Focus On

Core idea: Proportional reasoning connects algebra and geometry, enabling indirect measurement and scale problem solving.

Common stuck point: Set up proportions so corresponding parts align correctly—same position relative to equal angles in each ratio.

Sense of Study hint: Label each side of both figures by the angle it is opposite. Then match sides that face equal angles when writing your proportion.

Worked Examples

Example 1

easy
Two similar triangles have corresponding sides in the ratio 3:5. If the shorter triangle has a base of 9 cm, what is the base of the larger triangle?

Solution

  1. 1
    Step 1: Set up the proportion: \dfrac{\text{short base}}{\text{long base}} = \dfrac{3}{5}.
  2. 2
    Step 2: \dfrac{9}{x} = \dfrac{3}{5}.
  3. 3
    Step 3: Cross-multiply: 3x = 45 \Rightarrow x = 15 cm.

Answer

15 cm
Similar figures have all corresponding lengths in the same ratio (the scale factor). Setting up a proportion and cross-multiplying is the standard method to find a missing length in similar figures.

Example 2

medium
A 6 ft tall person casts a shadow 4 ft long. At the same time, a tree casts a shadow 14 ft long. How tall is the tree?

Example 3

medium
Two similar rectangles have widths 4 cm and 10 cm. If the smaller rectangle has length 6 cm, find the length of the larger rectangle.

Practice Problems

Try these problems on your own first, then open the solution to compare your method.

Example 1

easy
Two similar rectangles have widths 5 cm and 15 cm. What is the scale factor from the smaller to the larger?

Example 2

hard
On a map, 1 cm represents 50 km. Two cities are 3.6 cm apart on the map. What is the actual distance? Also, a lake has an area of 4 cm² on the map. What is its actual area in km²?
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Background Knowledge

These ideas may be useful before you work through the harder examples.

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