Library / /179 of 1394

, 2016

Geometry Difficulty 4.8 AIME Prove it United States

Problem:

Let $z$ be a complex number such that $|z|=1$ and $|z-1.45|=1.05$ . Compute the real part of $z$ .

Solution

Solution:

From the problem, let $A$ denote the point $z$ on the unit circle, $B$ denote the point $1.45$ on the real axis, and $O$ the origin. Let $A H$ be the height of the triangle $O A H$ and $H$ lies on the segment $O B$ . The real part of $z$ is $O H$ . Now we have $O A = 1$ , $O B = 1.45$ , and $A B = 1.05$ . Thus
$O H = O A \cos \angle A O B = \cos \angle A O B = \frac{1^2 + 1.45^2 - 1.05^2}{2 \cdot 1 \cdot 1.45} = \frac{20}{29}$

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Source: MathNet, licensed CC-BY-4.0. Statement reproduced verbatim; metadata (topic, difficulty) added by this project.