Proportional Geometry Math Example 5

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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? Also, a lake has an area of 4 cm² on the map. What is its actual area in km²?

Solution

  1. 1
    Step 1: Linear scale: 1 cm = 50 km, so 3.6 cm = 3.6×50=1803.6 \times 50 = 180 km.
  2. 2
    Step 2: Area scale factor = (50)2=2500(50)^2 = 2500 km²/cm².
  3. 3
    Step 3: Lake area = 4×2500=10,0004 \times 2500 = 10{,}000 km².

Answer

Distance: 180 km. Lake area: 10,000 km².
Map scales are linear ratios. For areas, you must square the linear scale factor because area is two-dimensional. This is a direct application of the scaling law: lengths scale by kk, areas by k2k^2.

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 2 medium

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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

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