AMC 10 · 2012 · #21

Grade 8 geometry-3d
Archetype Coordinate Plane Figure Computation └ Sub-archetype Pythagorean / Distance Formula for a Length
coordinate-geometryarea-rectanglespythagorean-theorem physical-representationidentify-subproblems ↑ Prerequisites: coordinate-geometrypythagorean-theorem
📏 Medium solution 💡 3 insights
Problem
Four points A=(0,0,0), B=(1,0,0), C=(0,2,0), and D=(0,0,3) sit in 3D space. E, F, G, H are the midpoints of the segments BD, AB, AC, DC, in that order. Find the area of the quadrilateral EFGH.

Pick an answer.

(A)
$\sqrt{2}$
(B)
$\frac{2\sqrt{5}}{3}$
(C)
$\frac{3\sqrt{5}}{4}$
(D)
$\sqrt{3}$
(E)
$\frac{2\sqrt{7}}{3}$

AMC 10 2012 problem © Mathematical Association of America (MAA AMC). Reproduced for educational use.

How to solve
Strategy Draw a Diagram

Put every point on the coordinate grid. Once the four midpoints have real coordinates, the shape of EFGH stops being a mystery: I can break the job into small subproblems (find the corners, check the sides, then measure them) and use spatial reasoning to see which sides meet at a right angle.

1STEP 1

Find the four midpoints

Average each pair of endpoints coordinate by coordinate: E=(1/2,0,3/2), F=(1/2,0,0), G=(0,1,0), H=(0,1,3/2).

E=((1+0)/2,(0+0)/2,(0+3)/2)=(1/2,0,3/2), F=(1/2,0,0), G=(0,1,0), H=(0,1,3/2)
2STEP 2

Check opposite sides

EF moves (0,0,-3/2), GH moves (0,0,3/2); FG moves (-1/2,1,0), HE moves (1/2,-1,0). Opposite sides match: EFGH is a parallelogram.

EF=F-E=(0,0,-3/2), GH=H-G=(0,0,3/2); FG=G-F=(-1/2,1,0), HE=E-H=(1/2,-1,0)
3STEP 3

Find the right angle

EF runs straight up the z-axis while FG lies flat in the xy-plane, so they meet at a square corner: the parallelogram is a rectangle.

EF · FG = (0)(-1/2)+(0)(1)+(-3/2)(0)=0
4STEP 4

Measure the two sides

The 3D distance formula on those shifts gives |EF|=3/2 and |FG|=√(1/4+1)=√5/2.

|EF|=√(0²+0²+(3/2)²)=3/2, |FG|=√((1/2)²+1²+0²)=√(5/4)=√5/2
5STEP 5

Compute the area

A rectangle's area is length times width: 3/2 · √5/2 = 3√5/4, which is choice (C).

Area=|EF|·|FG|=3/2·√5/2=3√5/4
Answer
3√(5)/4
Numerically 3√5/4≈ 1.68, which sits sensibly between sides of length 1.5 and about 1.12. The right-angle check (EF · FG=0) and the equal opposite sides both confirm a genuine rectangle, so length times width is the correct area rule. Answer (C) holds.
💡Key takeaway

Give every point coordinates, average them to find midpoints, and the hidden shape turns out to be a plain rectangle you can just measure.

  • Find the four midpoints
  • Check opposite sides
  • Find the right angle
  • Measure the two sides
  • Compute the area

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