HMMT 十一月 2023 · 团队赛 · 第 3 题

HMMT November 2023 — Team Round — Problem 3

[25] Two distinct similar rhombi share a diagonal. The smaller rhombus has area 1 , and the larger rhombus has area 9 . Compute the side length of the larger rhombus.

解析

[25] Two distinct similar rhombi share a diagonal. The smaller rhombus has area 1, and the larger rhombus has area 9. Compute the side length of the larger rhombus. Proposed by: Albert Wang √ Answer: 15 Solution: Let d be the length of the smaller diagonal of the smaller rhombus. Since the ratio of the areas is 9 : 1, the ratio of the lengths is 3 : 1. This means that the smaller diagonal of the larger rhombus (which is also the longer diagonal of the smaller rhombus) has length 3 d . 3 d 2 d 2 (Diagram not to scale.) Therefore, the smaller rhombus has diagonal lengths d and 3 d , so since it has area 1, we have r 1 2 d (3 d ) = 1 = ⇒ d = . 2 3 By the Pythagorean Theorem, the side length of the smaller rhombus is s r r 2 2 2 d 3 d 5 d 5

= = . 2 2 2 3 √ The side length of the larger rhombus is three times this, i.e. 15 .