Track / Stage 5 / 11 of 400 #611 of 2000

Problem 611

AIME late

Geometry Difficulty 5.0 Find the answer

4. The diagram shows three congruent regular hexagons. Some diagonals have been drawn, and some regions then shaded. The total shaded areas of the hexagons are $X, Y, Z$ as shown. Which of the following statements is true?

A $X, Y$ and $Z$ are all the same
B $Y$ and $Z$ are equal, but $X$ is different
C $X$ and $Z$ are equal, but $Y$ is different
D $X$ and $Y$ are equal, but $Z$ is different
E $\quad X, Y, Z$ are all different

Your answer

Official solution

4. A By joining the vertices of the inner triangle to the centre of the hexagon in both the first and third diagrams, it can be seen that each hexagon has been dissected into six equal parts, three of which are shaded. Therefore, $X, Y$ and $Z$ are each half of the hexagon and hence they are all the same.

Source: NuminaMath-1.5, licensed Apache-2.0. Statement and solution reproduced as published; topic, difficulty and ordering added by this site.