AMC 10 · 2007 · #13

Grade 9 geometry-2dalgebra

Archetype Coordinate Plane Figure Computation └ Sub-archetype Line Intersection Point as the Answer

coordinate-geometrycompleting-the-squareoptimizationperpendicular-slopes convert-to-algebracompleting-the-square ↑ Prerequisites: coordinate-geometry

📏 Medium solution 💡 3 insights

Problem

A mouse runs along a straight line and at some point stops closing in on a fixed target and begins moving away. Find the sum of the coordinates where that switch happens.

Pick an answer.

(A)

(B)

(C)

(D)

(E)

How to solve

Strategy Extreme Principle

The switch from closing in to falling behind happens exactly where the distance is smallest, so Tool #14 (Extreme Principle) is the spine of the problem. To find that minimum, Tool #4 (Introduce a Variable) collapses the mouse's whole path onto a single number x, since a point on the line is fully described by its x-coordinate. Tool #15 (Organize Information in More Ways) then swaps distance for squared distance — the square root only hides the algebra, and comparing squares ranks distances the same way — which turns the question into a plain quadratic whose vertex is visible after completing the square. Tool #1 (Draw a Diagram) carries the piece every quick solution skips: which way along the line the mouse is actually running. The minimum of the quadratic sits at x=2 and the mouse starts at x=4, so the direction of travel decides whether the mouse ever reaches that point at all.

1STEP 1

Name the mouse's position

One parameter names the runner's position along the line.

P(x)=(x, 18-5x), P(4)=(4,-2)

2STEP 2

Which way is "up"?

Checking the direction of travel matters before anything else.

y=18-5x increasing ⇔ x decreasing; x: 4 → 3 → 2 → …

3STEP 3

Measure with squared distance

Using the squared distance avoids the root entirely.

D(x)=(x-12)²+(8-5x)²=(x²-24x+144)+(25x²-80x+64)=26x²-104x+208

4STEP 4

Complete the square

Completing the square finds where it bottoms out.

D(x)=26((x-2)²+4), D(2)=104

5STEP 5

Match the valley to the trip

That valley falls inside the trip, so it really is the switch.

x:4↓ 2 → closer; x ＜ 2 → farther; (a,b)=(2,8), a+b=10

6STEP 6

Confirm with a right angle

A right angle confirms it, so the sum is 10, choice (B).

(C-F)·(1,-5)=(10)(1)+(2)(-5)=0; CP²=CF²+FP²=104+26(x-2)² → a+b=10 → (B)

Answer

The five choices can all be tested at once. Since (a,b) is on the line, b=-5a+18, so a+b=-4a+18; each choice therefore pins down one candidate point. (A) 6→ a=3, point (3,3); (B) 10→ a=2, point (2,8); (C) 14→ a=1, point (1,13); (D) 18→ a=0, point (0,18); (E) 22→ a=-1, point (-1,23). Their squared distances to (12,10) are 130, 104, 130, 208, 338. Only (B) is the smallest, and the tie between (A) and (C) at 130 is the expected symmetry of a parabola about x=2. Two more sanity checks pass: the mouse's starting squared distance is (4-12)²+(-2-10)²=208, larger than 104, so it genuinely does close in; and 208 also matches the value at x=0, exactly as the symmetry about x=2 demands. The closest approach √(104)=2√(26)≈ 10.2 is comfortably less than the starting √(208)≈ 14.4.

💡Key takeaway

Squared distance along a straight path is a parabola with one bottom, so the moment you start losing ground is the point where your path meets the target at a right angle — provided you were running toward it in the first place.

Name the mouse's position
Which way is "up"?
Measure with squared distance
Complete the square
Match the valley to the trip
Confirm with a right angle