# 10.3.3. The Relation between the Area of Two Similar Polygons

In , where , and where . If the area of , determine the area of the trapezium .

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### Example

In , where , and where . If the area of , determine the area of the trapezium .

### Solution

If a line is drawn parallel to one side of a triangle and intersects the other two sides or the lines containing them, then the resulting triangle is similar to the original triangle. This means that and are similar. In and , side corresponds to side , side corresponds to side , and side corresponds to side . First, use the fact that to find the length of as follows:

Since the length of is , the length of is . Now use the fact that the ratio of the areas of the surfaces of two similar triangles equals the square of the ratio of the lengths of any two corresponding sides of the triangles to find the area of .

Since the area of trapezium is equal to the area of minus the area of , the area of trapezium is equal to .

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