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This problem trains for: SAT-I, AMC-8, GMAT, AMC-10.
The midpoint M of a segment AB has coordinates (5,0). The segment AB makes an angle of 30° with the positive direction of the x-axis. If the equilateral triangle ABC intersects the x-axis at point D(2,0), what is the area of the triangle?
Solution 1: Knowing that ∠DMA=30° and that ∠DAM=60°, we figure out that the triangle DMA is a 30°-60°-90° triangle.
From the coordinates of the points D and M we infer that the segment DM has length 3:
Multiply the right hand ratio by √3 in order to obtain the needed 3:
and the area of the triangle is:
and find CM by applying again the special ratios:
Solution 2: It is easier to solve using the similarity of triangles:
Since M is the midpoint, the similarity ratio is 2:1.
Then, the areas are in a ratio of 4:1.
And since the triangle CMB is the half of ABC then it is sufficient to compute the area of DMA and multiply it by 8.