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

Find the area of the triangle MNP if the large square has side length 4 units, the small squares have side length 2 units and the smallest angles formed by the sides of the small squares with the sides of the large square have measure 45°.

figure

The triangles OAM and OBN are congruent where M, N and O are the centers of the three squares.

figure

The length of the segment AM is:

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By the Pythagorean theorem, the length of the segment OM is:

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By chasing the angle measures in the figure, we obtain:

equation

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Since, in the right angle triangles OAM and OBM we have:

equation

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it follows that these triangles are congruent:

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and

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Therefore the triangle OMN is also a right angle triangle. Moreover, since the triangles OAM and OBM are congruent, it is an isosceles right triangle. Its area is:

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