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Geometry Difficulty 4.1 AIME Find the answer United States

Problem:

Arpon chooses a positive real number $k$ . For each positive integer $n$ , he places a marker at the point $(n, n k)$ in the $(x, y)$ plane. Suppose that two markers whose $x$ coordinates differ by $4$ have distance $31$ . What is the distance between the markers at $(7,7 k)$ and $(19,19 k)$ ?

Your answer

A number or a short expression. Fractions can be typed as 3/2, and spacing doesn't matter.

Solution

Solution:

The difference of the $x$ -coordinates of the markers is $12 = 3 \cdot 4$ . Thus, by similar triangles (where we draw right triangles whose legs are parallel to the axes and whose hypotenuses lie along the line $y = k x$ ), the distance between the markers is $3 \cdot 31 = 93$ .

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