Proportional Geometry Math Example 2

Follow the full solution, then compare it with the other examples linked below.

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?

Solution

  1. 1
    Step 1: The sun's rays form similar triangles — person and shadow, tree and shadow.
  2. 2
    Step 2: Set up proportion: person heightperson shadow=tree heighttree shadow\dfrac{\text{person height}}{\text{person shadow}} = \dfrac{\text{tree height}}{\text{tree shadow}}.
  3. 3
    Step 3: 64=h14\dfrac{6}{4} = \dfrac{h}{14}.
  4. 4
    Step 4: h=6×144=844=21h = \dfrac{6 \times 14}{4} = \dfrac{84}{4} = 21 ft.

Answer

The tree is 21 ft tall.
This classic similar-triangles application uses the fact that parallel sun rays create triangles with the same angles. The ratio of height to shadow length is constant for both objects at the same time. This technique was used by ancient Greek mathematicians to measure pyramid heights.

About Proportional Geometry

Proportional geometry studies how corresponding lengths, areas, and volumes scale between similar figures. If two triangles are similar with scale factor k, their sides are in ratio k, their areas in ratio k², and their volumes in ratio k³.

Learn more about Proportional Geometry →

More Proportional Geometry Examples

Example 1 easy

Two similar triangles have corresponding sides in the ratio 3:5. If the shorter triangle has a base

Example 3 medium

Two similar rectangles have widths [formula] cm and [formula] cm. If the smaller rectangle has lengt

Example 4 easy

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

Example 5 hard

On a map, 1 cm represents 50 km. Two cities are 3.6 cm apart on the map. What is the actual distance

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