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11

Part of 1988 AIME Problems

Problems(1)

Mean line

Source:

3/19/2006
Let w1,w2,…,wnw_1, w_2, \dots, w_n be complex numbers. A line LL in the complex plane is called a mean line for the points w1,w2,…,wnw_1, w_2, \dots, w_n if LL contains points (complex numbers) z1,z2,…,znz_1, z_2, \dots, z_n such that ∑k=1n(zk−wk)=0. \sum_{k = 1}^n (z_k - w_k) = 0. For the numbers w1=32+170iw_1 = 32 + 170i, w2=−7+64iw_2 = -7 + 64i, w3=−9+200iw_3 = -9 +200i, w4=1+27iw_4 = 1 + 27i, and w5=−14+43iw_5 = -14 + 43i, there is a unique mean line with yy-intercept 3. Find the slope of this mean line.
analytic geometrygraphing linesslopegeometrygeometric transformationratiocomplex numbers