2021 AMC Fall 10A Problem 17

P16-20~5 minPrint

Solid Geometry Angles & Polygons Coordinate Geometry Add construction lines/points (drop altitudes, extend segments, connect centers)Exploit symmetry to reduce work or pair up objects

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 $E$ ?

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Key insight

On a flat panel, equal steps give equal height changes. B to A adds 3, so the center is 13; center to E adds 4, so E is 17.

Solution

The key fact. The panel is flat. On a flat tilted surface, if you take the same step across the ground (same length, same direction) from two different starting points, the height changes by the same amount both times. Think of walking on a ramp: one step north always climbs the same amount.

Let $O$ be the center of the hexagon.

1. A useful step. A regular hexagon is made of six equilateral triangles meeting at $O$ . So $OA$ , $AB$ , $BC$ , $CO$ all have the same length, and $OABC$ is a rhombus. In a rhombus, opposite sides are parallel and equal, so the step from $B$ to $A$ is the same as the step from $C$ to $O$ .
2. Height at the center. From $B$ to $A$ the height goes from $9$ to $12$ , a change of $+3$ . The same step from $C$ to $O$ also changes the height by $+3$ :
$h(O) = 10 + 3 = 13.$
3. Height at $E$ . $B$ and $E$ are opposite corners, and $O$ is exactly halfway between them. So the step from $B$ to $O$ is the same as the step from $O$ to $E$ . From $B$ to $O$ the height goes from $9$ to $13$ , a change of $+4$ . So
$h(E) = 13 + 4 = 17.$

The answer is $\boxed{\textbf{(D)}\ 17}$ .

Why this works

A flat panel means height changes at a steady rate in every direction. So the height above a midpoint is the average of the heights at the two ends, and equal steps give equal changes. The center of a regular hexagon connects everything: it is halfway between each pair of opposite corners, and it forms a rhombus with any three corners in a row. Three known heights give the center, and the center gives the opposite corners.

Alternative approach

Use coordinates. Take side length $2$ , with $A = (2, 0)$ , $B = (1, \sqrt3)$ , $C = (-1, \sqrt3)$ , $E = (-1, -\sqrt3)$ , and write the panel's height as $h = ax + by + c$ .
- $A$ : $2a + c = 12$ .
- $B$ : $a + \sqrt3\, b + c = 9$ .
- $C$ : $-a + \sqrt3\, b + c = 10$ .

Subtracting the $C$ equation from the $B$ equation gives $2a = -1$ , so $a = -\frac12$ . Then $c = 13$ from the $A$ equation, and $\sqrt3\, b = 9 - 13 + \frac12 = -\frac72$ . At $E$ : $h = -a - \sqrt3\, b + c = \frac12 + \frac72 + 13 = 17$ .

Concise solution

A shorter write-up, the way contest solutions are usually written. It may take a different route from the solution above.

Because the panel is a flat plane, the height $h(P)$ above a ground point $P$ is a linear (affine) function of the position of $P$ . Two consequences we will use: the height above the midpoint of a segment is the average of the heights at its endpoints, and for any parallelogram $PQRS$ (where $\vec P + \vec R = \vec Q + \vec S$ ) we have $h(P) + h(R) = h(Q) + h(S)$ .

Let $O$ be the center of the hexagon. In a regular hexagon, $OA = AB = BC = CO$ and $OA \parallel CB$ , so $OABC$ is a rhombus, a parallelogram with opposite vertices $O, B$ and $A, C$ . Hence
$h(O) = h(A) + h(C) - h(B) = 12 + 10 - 9 = 13.$

Next, $O$ is the midpoint of the long diagonal $BE$ , so $h(O) = \dfrac{h(B) + h(E)}{2}$ , giving
$h(E) = 2 \cdot 13 - 9 = 17.$

The answer is $\boxed{\textbf{(D)}\ 17}$ .

Review in the AoPS books

Introduction to Geometry, Chapter 14: Three-Dimensional Geometry
Introduction to Geometry, Chapter 2: Angles
Introduction to Geometry, Chapter 17: Analytic Geometry
Introduction to Geometry, Chapter 19: Problem Solving Strategies in Geometry (§19.1)