AMC 10 · 2016 · #20

Grade 8 geometry-2d
Archetype Coordinate Plane Figure Computation └ Sub-archetype Figure Constraint Equations for Unknown Quantity
dilationcoordinate-geometryratio-proportion convert-to-algebra ↑ Prerequisites: coordinate-geometry
📏 Medium solution 💡 3 insights
Problem
A dilation (a size transformation with a positive scale factor) sends the circle of radius 2 centered at A(2,2) to the circle of radius 3 centered at A'(5,6). The same dilation moves every point of the plane, including the origin O(0,0). Find how far the origin moves to its image.

Pick an answer.

(A)
0
(B)
3
(C)
$\sqrt{13}$
(D)
4
(E)
5

AMC 10 2016 problem © Mathematical Association of America (MAA AMC). Reproduced for educational use.

How to solve
Strategy Introduce a Variable

A dilation is pinned down by exactly two facts: its scale factor and its center. The scale factor is free — it is just the ratio of the radii, 3/2. The center is hidden, so tool #4 (Introduce a Variable) names it C(h,j) and uses the one point we can track, A→ A', to write equations for h and j. Tool #13 (Convert to Algebra) turns the geometric rule "image point minus center equals scale factor times original point minus center" into linear equations. Tool #7 (Identify Subproblems) keeps the work in order: first the scale factor, then the center, then the image of the origin, then the distance. Tool #1 (Draw a Diagram) anchors which way the points sit so the algebra stays believable.

1STEP 1

Read off the scale factor

A dilation scales every length by one factor k, so the radii alone fix it: k = 3/2.

k = (new radius)/(old radius) = 3/2
2STEP 2

Name the center, write its equations

Let the center be C(h,j). A dilation obeys P' - C = k(P - C); apply it to A(2,2)→A'(5,6) in x and y with k = 3/2.

5 - h = 3/2(2 - h), 6 - j = 3/2(2 - j)
3STEP 3

Solve for the center

Expanding each equation collapses it to one variable: h = -4 and j = -6, so the center is C(-4,-6).

h = -4, j = -6 → C(-4,-6)
4STEP 4

Send the origin through the dilation

Apply O' = C + k(O - C) to the origin: center-to-O is (4,6), times 3/2 is (6,9), plus the center gives O'(2,3).

O' = (-4,-6) + 3/2(4,6) = (-4,-6) + (6,9) = (2,3)
5STEP 5

Measure how far the origin moved

The origin went from O(0,0) to O'(2,3); by the distance formula OO' = √(2² + 3²) = √(13), choice (C).

OO' = √((2-0)² + (3-0)²) = √(4+9) = √(13) = (C)
Answer
√(13)
Confirm the center actually works on the known pair: A' = C + k(A - C) = (-4,-6) + 3/2((2,2)-(-4,-6)) = (-4,-6) + 3/2(6,8) = (-4,-6) + (9,12) = (5,6) — exactly the given A', so the scale factor and center are consistent. The answer √(13) ≈ 3.6 sits between 3 and 4, both of which are decoy choices, and it is a clean radical matching choice (C). The origin moving a modest few units is sensible because the origin is not far from the center of dilation.
💡Key takeaway

A dilation is just a center plus a stretch factor: find both from the one pair of points you know, then push the origin through the same stretch and measure the gap.

  • Read off the scale factor
  • Name the center, write its equations
  • Solve for the center
  • Send the origin through the dilation
  • Measure how far the origin moved

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