Track / Stage 4 / 117 of 340 #377 of 1964

Problem 377

AMC 12 late, AIME early

Geometry Difficulty 4.7 Find the answer

A homothety is defined by its center $S$ and coefficient $k$ . Write the formula connecting $\vec{A}$ and $\vec{A}^{\prime}$ , where $A^{\prime}$ is the image of point $A$ under this homothety. (Note that the answer does not depend on the choice of origin O.)

Your answer

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

Official solution

13. If a homothety with center $S$ and coefficient $k$ maps point $A$ to $A^{\prime}$ , then $\left(\vec{A}^{\prime}-\vec{S}\right)=k(\vec{A}-\vec{S})$ , from which:

$\vec{A}^{\prime}=\overrightarrow{k A}+(1-k) \vec{S}$

Source: NuminaMath-1.5, licensed Apache-2.0. Statement reproduced verbatim; metadata (topic, difficulty, ordering) added by this project.