Equal-Area Transformations (Changing the Shape Without Changing the Area)
“Equal-area transformation” is a technique for changing the shape of a figure while keeping its area unchanged. It is used to transform a difficult figure into one that is easier to calculate.
Rules for Equal-Area Transformations
- If the bases and heights are equal, the areas are equal
- Even if the vertex of a triangle is moved along a line parallel to the base, its area does not change
- Divide a difficult polygon into several triangles and find the area
Example 1: Point P lies on diagonal BD of parallelogram ABCD. What fraction of the area of the parallelogram is the sum of the areas of △ABP and △CDP?
Solution:
① △ABD is exactly half of the parallelogram (1/2)
② If P lies on BD, △ABP + △CDP = △ABD
③ Therefore, the answer is 1/2 of the parallelogram
Solution:
① △ABD is exactly half of the parallelogram (1/2)
② If P lies on BD, △ABP + △CDP = △ABD
③ Therefore, the answer is 1/2 of the parallelogram
Example 2: Change only the shape of a triangle with a base of 8 cm and a height of 5 cm, keeping the base and height the same. What is the area of the new triangle?
Solution: Area = (1/2) × 8 × 5 = 20 cm² (the area does not change even if the shape changes)
Solution: Area = (1/2) × 8 × 5 = 20 cm² (the area does not change even if the shape changes)
The key to equal-area transformations is finding figures with “the same base and height.” Practice the idea of drawing parallel lines and sliding the vertex.