AMC 10 · 2008 · #19

Grade 8 geometry-2d

Archetype Path Length Comparison & Shortest Path └ Sub-archetype Sum Known Path Segments

rotation-isometrypythagorean-theoremcoordinate-geometry spatial-visualizationphysical-representation ↑ Prerequisites: pythagorean-theorem

📏 Long solution 💡 3 insights

Problem

Rectangle PQRS lies in a plane with PQ = RS = 2 and QR = SP = 6. The rectangle is first rotated 90 degrees clockwise about R, then rotated 90 degrees clockwise about the point that S moved to after the first rotation. What is the length of the path traveled by point P?

Pick an answer.

(A)

$\ \left(2\sqrt{3}+\sqrt{5}\right)\pi$

(B)

$\ 6\pi$

(C)

$\ \left(3+\sqrt{10}\right)\pi$

(D)

$\ \left(\sqrt{3}+2\sqrt{5}\right)\pi$

(E)

$\ 2\sqrt{10}\pi$

AMC 10 2008 problem © Mathematical Association of America (MAA AMC). Reproduced for educational use.

How to solve

Strategy Visualize Spatial Relationships

The whole problem is tracking one corner through two rotations, so mentally (and on a grid) picturing where each point goes is the core skill. Coordinates pin the rotations down exactly, and the total path splits neatly into two separate arcs, each a quarter-circle whose radius is just P's distance from the pivot.

1STEP 1

Put the rectangle on a grid

Place P at (0,6), Q at (2,6), R at (2,0), S at (0,0) — the sides then measure 2 and 6, matching the givens.

P=(0,6), Q=(2,6), R=(2,0), S=(0,0)

2STEP 2

Find the first arc's radius

The first turn spins about R, so P keeps its distance from R — the arc radius, from legs 2 and 6: 2√(10).

PR=√(2²+6²)=√(40)=2√(10)

3STEP 3

Length of the first quarter-circle

A 90-degree turn is a quarter circle, so P sweeps a quarter of the circumference of radius 2√(10): √(10) π.

1/4 · 2π · 2√(10)=√(10) π

4STEP 4

Track where S and P land

A clockwise quarter-turn about R sends (x,y) to (2+y, 2-x), so S lands at (2,2) and P at (8,2).

(x,y)↦(2+y, 2-x): S→(2,2), P→(8,2)

5STEP 5

Length of the second quarter-circle

P at (8,2) turns about (2,2); same height, so the radius is 6 and the quarter-arc is 3π.

radius=6, 1/4 · 2π · 6=3π

6STEP 6

Add the two arcs

The two arcs join with no gap, so add them: √(10) π + 3π = (3+√(10))π, choice (C).

√(10) π+3π=(3+√(10))π

Answer

(3+√(10))π

The two radii, 2√(10)≈6.3 and 6, are both close to 6, and each contributes a quarter-circle, so the total should be a bit above the two quarter-circumferences of a radius-6 circle: roughly √(10) π+3π≈3.16π+3π=6.16π. Choice (C) equals (3+√(10))π≈6.16π, which fits. Choice (B) 6π is close but ignores that the first radius exceeds 6, so (C) is right.

💡Key takeaway

Each 90-degree turn sends a corner along a quarter-circle whose radius is just its distance from the pivot, so add the two quarter-arcs to get the whole path.

Put the rectangle on a grid
Find the first arc's radius
Length of the first quarter-circle
Track where S and P land
Length of the second quarter-circle
Add the two arcs