2021 AMC 10A Fall Problem 17

Question

An architect is building a structure that will place vertical pillars at the vertices of regular hexagon

ABCDEF,

which is lying horizontally on the ground. The six pillars will hold up a flat solar panel that will not be parallel to the ground. The heights of pillars at

A,

B,

and

C

are

12,

9,

and

10

meters, respectively. What is the height, in meters, of the pillar at

E?

9

6\sqrt{3}

8\sqrt{3}

17

12\sqrt{3}

Answer: D

Concepts:coordinate geometry regular polygon system of equations

Difficulty rating: 1660

Small hint:

A flat panel means height is linear over the ground plane

Big hint:

Use regular-hexagon coordinates and a function

ux+vy+w

Solution:

Put a regular hexagon in coordinates with

A=(-1,0),

B=(-\frac{1}{2},\frac{\sqrt3}{2}),

C=(\frac{1}{2},\frac{\sqrt3}{2}),

and

E=(\frac{1}{2},-\frac{\sqrt3}{2}).

Because the solar panel is flat, the height is an affine function

h(x,y)=ux+vy+w.

From

h(A)=12,

h(B)=9,

and

h(C)=10,

subtracting the last two equations gives

u=1.

Then

-u+w=12,

so

w=13.

Using

h(B)=9

gives

-\frac{1}{2}+\frac{\sqrt3}{2}v+13=9,

so

\sqrt3v=-7.

Therefore

\begin{aligned} h(E) &=\frac{1}{2}-\frac{\sqrt3}{2}v+13 \\ &=\frac{1}{2}+\frac{7}{2}+13 \\ &=17. \end{aligned}

Thus, D is the correct answer.