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SOURCE:COMPETITION Number of Problems: 26. FOR PRINT ::: (Book)
An equiangular octagon has four sides of length 1 and four sides of length , arranged so that no two consecutive sides have the same length. What is the area of the octagon?
The area of the octagon can be divided up into 5 squares with side and 4 right triangles, which are half the area of each of the squares.
Therefore, the area of the octagon is equal to the area of squares.
The area of each square is , so the area of 7 squares is .
In we have , , and . Points and are on and respectively, with and . What is the ratio of the area of triangle to the area of the quadrilateral ?
The area of a triangle is .
Using this formula:
Since the area of is equal to the area of minus the area of ,
.
Therefore, the desired ratio is
In rectangle , points and trisect , and points and trisect . In addition, . What is the area of quadrilateral shown in the figure?
It is not difficult to see by symmetry that is a square. Draw . Clearly . Then is isosceles, and is a . Hence , and .
There are many different similar ways to come to the same conclusion using different 45-45-90 triangles.
Draw the lines as shown above, and count the squares. There are 12, so we have .
A rectangular parking lot has a diagonal of meters and an area of square meters. In meters, what is the perimeter of the parking lot?