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This problem trains for: SAT-II, AMC-10, Math Kangaroo 7-8, Math Kangaroo 9-10.

A circle of radius 10 is reflected across a chord of length 10√3. What is the area of the figure thus formed?

figure

According to the inclusion-exclusion principle the area formed can be calculated as:

equation

The area of the intersection can be further computed as the double of the 'moon' shape formed between the chord and the circular boundary:

equation

equation

Thus:

equation

We have to compute the area of the triangle and the area of the sector.

The triangle is formed by the given chord and the two radii that connect its ends to the center of a circle. This is an isosceles triangle with base 10√3 and equal sides of length 10.

Draw an altitude along its line of symmetry and obtain a right angle triangle with sides 5√3 and 10 which are in a ratio of √3 : 2, indicating a 30-60-90 triangle.

Therefore, the altitude is 5 units long and the area is:

equation

The 60° angle of this triangle has its vertex at the center of the circle. Therefore the sector of interest has central angle 120°. Its area is one third of the area of the circle:

equation

The area required is:

equation