Rectangle PQRS lies in a plane with PQ=RS=2 and QR=SP=6. The rectangle is rotated 90∘ clockwise about R, then rotated 90∘ clockwise about the point S moved to after the first rotation. What is the length of the path traveled by point P?
Try it yourself first — the explanation is most useful after you’ve attempted it.
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Toolkit + CCSS Solution
Understand
Restated: A 2-by-6 rectangle PQRS is turned a quarter-turn clockwise about corner R, then a quarter-turn clockwise about the spot where S landed. Find the total length of the curved path that corner P traces during both turns.
Givens: Rectangle PQRS has PQ = RS = 2 and QR = SP = 6.; First move: rotate the rectangle 90 degrees clockwise about R.; Second move: rotate 90 degrees clockwise about the new position of S (after the first rotation).
Unknowns: The total length of the path traveled by point P across both rotations.
Understand
Restated: A 2-by-6 rectangle PQRS is turned a quarter-turn clockwise about corner R, then a quarter-turn clockwise about the spot where S landed. Find the total length of the curved path that corner P traces during both turns.
Givens: Rectangle PQRS has PQ = RS = 2 and QR = SP = 6.; First move: rotate the rectangle 90 degrees clockwise about R.; Second move: rotate 90 degrees clockwise about the new position of S (after the first rotation).
Plan
Primary tool: #17 Visualize Spatial Relationships
Secondary: #1 Draw a Diagram, #7 Identify Subproblems, #4 Introduce a Variable
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.
Execute — Answer: C
#1 Draw a Diagram 6.NS.C.8Step 1
Put the rectangle on a grid
Give each corner coordinates so the turns become arithmetic.
Place P at (0,6), Q at (2,6), R at (2,0), and S at (0,0).
Then PQ = 2, QR = 6, RS = 2, and SP = 6, which matches the given side lengths.
$$P=(0,6),\ Q=(2,6),\ R=(2,0),\ S=(0,0)$$
💡 Coordinates turn 'rotate about a corner' into distances you can compute.
#17 Visualize Spatial Relationships 8.G.B.8Step 2
Find the first arc's radius
In the first rotation everything spins around R, so P stays exactly its starting distance from R.
That distance is the radius of P's circular path.
Use the right triangle with legs 2 and 6.
$$PR=\sqrt{2^2+6^2}=\sqrt{40}=2\sqrt{10}$$
💡 A rotation keeps every point the same distance from the pivot, so that distance is the arc's radius.
#7 Identify Subproblems 7.G.B.4Step 3
Length of the first quarter-circle
A 90-degree turn is one quarter of a full 360-degree circle, so P sweeps one quarter of a circle of radius 2\sqrt{10}.
💡 The path is unbroken, so its total length is just the sum of the two arc lengths.
[1]
#1 6.NS.C.8Give each corner coordinates so the turns become arithmetic. Place P at (0,6), Q
[2]
#17 8.G.B.8In the first rotation everything spins around R, so P stays exactly its starting
[3]
#7 7.G.B.4A 90-degree turn is one quarter of a full 360-degree circle, so P sweeps one qua
[4]
#17 8.G.A.3The second pivot is where S goes after the first turn, so follow the corners. A
[5]
#7 7.G.B.4Now P at (8,2) rotates 90 degrees about the new pivot (2,2). Both points share t
[6]
#7 8.EE.A.2The path is the two arcs joined end to end with no gap, so add their lengths. \s
Review
Reasonableness: The two radii, 2\sqrt{10}\approx6.3 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 \sqrt{10}\,\pi+3\pi\approx3.16\pi+3\pi=6.16\pi. Choice (C) equals (3+\sqrt{10})\pi\approx6.16\pi, which fits. Choice (B) 6\pi is close but ignores that the first radius exceeds 6, so (C) is right.
Alternative: Instead of coordinates, use the Pythagorean theorem directly on the rectangle: P's distance to R is the diagonal of a 2-by-6 rectangle, \sqrt{40}=2\sqrt{10}; and after the first flip, P's distance to the moved S is the long side, 6. Two quarter-circles of those radii give \tfrac{1}{4}(2\pi)(2\sqrt{10})+\tfrac{1}{4}(2\pi)(6)=(3+\sqrt{10})\pi, the same answer without a grid.
CCSS standards used (min grade 8)
6.NS.C.8 Solve real-world problems by graphing points in all four quadrants (Placing the rectangle's corners on a coordinate grid to make the rotations computable.)
8.G.B.8 Apply the Pythagorean theorem to find distance between two points in a coordinate system (Finding P's distance to pivot R, which is the radius of the first arc.)
7.G.B.4 Know the formulas for area and circumference of a circle (Computing each quarter-circle arc length from its radius.)
8.G.A.3 Describe the effect of dilations, translations, rotations, and reflections on coordinates (Mapping where S and P land after the first 90-degree clockwise rotation.)
8.EE.A.2 Use square root and cube root symbols to represent solutions (Keeping the total path in exact radical form (3+\sqrt{10})\pi.)
⭐ 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.
⭐ 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.
