Chapter 7 · Closed-route travel

Lesson 7.5 — Positions on Polygonal Closed Tracks

多边形封闭路线上的位置

A triangle, square, or rectangle may look different from a circle, but its perimeter is still one repeating route. Choose a starting coordinate, follow the directed sides, remove complete laps, and place the remainder back on the correct side.

Turn the perimeter into one repeating number line.

Grade 5 enrichment10 interactive missionsTriangle · square · rectangle · pentagonAutosaves in this browser

Mission 1

Give the perimeter coordinates

Choose a starting point and one direction. Label each vertex by distance along the perimeter from the start. Throughout this lesson, travelers start simultaneously and move continuously at constant stated speeds, without stopping at vertices. Meetings exclude time 0.

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An equilateral triangular track

The perimeter is 360 m, so every side is:

360 ÷ 3 = 120 m

Use the forward direction A → B → C → A.

120B

240C

360 = 0A again

Key idea: Coordinates 0 and 360 describe the same physical point A after one complete lap.

Closed track and unwrapped routeCore model

The straight strip below the triangle is the same perimeter cut open at A.

Side length (m)

Coordinate of B

Coordinate of C

Mission 2

Remove complete laps first

A long traveled distance may contain several full perimeters. Complete laps return the traveler to the same point, so the remainder determines the final position.

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Triangle position explorer

Distance traveled231 m

0complete laps

231remainder (m)

BCcurrent side

9metres to next vertex

The traveler is on BC, 111 m from B and 9 m from C.

distance = complete laps × perimeter + remainder

The remainder is always from 0 up to—but not including—the full perimeter.

Live triangle positionMove the slider

The marker on the triangle and the marker on the strip describe the same location.

Check with 510 m

On the 360-m triangle, a traveler moves 510 m from A in the direction A → B → C → A.

Complete laps

Remainder (m)

Current side

Distance from B (m)

Mission 3

Wrap the remainder back onto the polygon

A perimeter coordinate tells both the side and the distance from its first vertex. Do not replace route distance with a straight chord.

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Example: coordinate 330

Vertex C is at coordinate 240.

The next side is C → A, from coordinate 240 to 360.

330 − 240 = 90, so the traveler is 90 m from C.

360 − 330 = 30, so the traveler is 30 m from A.

Route distance: Here the traveler and C lie on side CA, so their straight segment is also part of the route. Do not use a diagonal through the polygon to replace a distance around its boundary.

Closed or unwrapped?

Both views describe the same coordinate 330 m.

Current side at coordinate 330

Distance from C (m)

Distance to A (m)

Mission 4

Use directed arcs to find the first meeting

This mission develops the chapter’s triangular-track example. The two travelers move in opposite directions, so use the route distance between them and add their speeds.

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The complete situation

Equilateral triangle perimeter: 360 m
Each side: 120 m
Traveler A starts at A and moves A → B at 55 m/min
Traveler B starts at D on BC, 30 m from C, and moves toward B at 50 m/min

Because D is 30 m from C, it is:

120 − 30 = 90 m from B

The directed gap is:

AB + DB = 120 + 90 = 210 m

The closing speed is:

55 + 50 = 105 m/min

So the meeting time is:

210 ÷ 105 = 2 min

original journey simulatorChapter example

Elapsed time0.00 min

0A coordinate

210B coordinate

210route gap (m)

Approachingstate

Directed starting gap (m)

Closing speed (m/min)

Meeting time (min)

Meeting point is how far from B on AB? (m)

Mission 5

Locate one traveler when the other reaches a vertex

Continue Mission 4: find traveler A’s position when traveler B, who started at D, first reaches vertex A. Use a time event first, then reduce A’s traveled distance to a perimeter coordinate.

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Build the event

B travels from D to B and then B to A: 90 + 120 = 210 m.

B’s time to A: 210 ÷ 50 = 4.2 min.

In 4.2 minutes, A travels: 55 × 4.2 = 231 m.

Coordinate 231 lies on BC. It is 240 − 231 = 9 m from C.

Two different roles: B’s movement determines the time. A’s movement determines the requested position.

Move time toward B’s arrival at AEvent at 4.2 min

Elapsed time4.2 min

B is at A. Traveler A is on BC, 9 m from C.

Traveler B’s distance from D to vertex A (m)

Time until traveler B first reaches vertex A (min)

A’s distance traveled (m)

A’s distance from C (m)

Mission 6

Use one locator for many polygonal tracks

The method does not depend on a circle. Select a track, travel in the listed vertex order, and let the remainder choose the final side.

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Track

Distance230 m

320perimeter

0full laps

230remainder

CDcurrent side

General polygon locatorInteractive model

The listed vertex order is the forward direction.

Square check

A square has side length 80 m. Starting at A, a traveler follows A → B → C → D → A for 230 m.

Perimeter (m)

Current side

Distance to D (m)

Mission 7

Use unequal side lengths on a rectangle

A rectangle’s vertex coordinates are not equally spaced unless it is a square. Accumulate the actual side lengths.

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Rectangle 120 m by 60 m

Follow A → B → C → D → A.

120B

180C

300D

The perimeter is:

120 + 60 + 120 + 60 = 360 m

After 410 m:

410 = 1 × 360 + 50

Coordinate 50 lies on AB, 50 m from A.

Rectangle position explorerUnequal intervals

Distance410 m

Rectangle perimeter (m)

Remainder after 410 m

Current side

Mission 8

Track which side each traveler occupies

This develops the square-wall exercise from the chapter. Use this idealized visibility rule: walkers follow the square wall’s boundary and can see each other when they lie on the same side segment, including its endpoints. The opaque wall blocks views across its interior.

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Square wall original extension

Side length: 300 m
The faster walker starts at A; the slower starts at opposite vertex C
Both move counterclockwise, following A→B→C→D→A
Faster walker: 90 m/min
Slower walker: 70 m/min

One side takes:

300 ÷ 90 = 10/3 min300 ÷ 70 = 30/7 min

Compare the side occupied after every vertex event. The first matching side occurs at:

50/3 min = 16⅔ min

At that instant, both are on side BC.

First visible positionSquare wall

The faster walker has just reached B; the slower walker is still on BC.

Side-event schedule

Time	Faster walker	Slower walker	Same side?
0	AB	CD	No
10/3	BC	CD	No
30/7	BC	DA	No
20/3	CD	DA	No
60/7	CD	AB	No
10	DA	AB	No
90/7	DA	BC	No
40/3	AB	BC	No
50/3	BC	BC	Yes

Earliest time (minutes; decimal (for example 16.67) or fraction (for example 50/3))

First shared side

Mission 9

Polygon-position workshop

All eight answers are numbers. In Questions 1, 4, 5, and 6, start at vertex A and follow the vertices in alphabetical order around the boundary. Questions 2–3 use Missions 4–5, Question 5 uses Mission 7’s rectangle, and Question 8 uses Mission 8’s wall model. Give distances in metres and times in minutes.

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1Triangle · 490 m

On the 360-m triangle, how far past B is the traveler?

2Mission 4 meeting

First meeting time, in minutes?

3Mission 5 position

When B reaches A, how far is A from C?

4Square · 275 m

On a 100-m-side square, how far is the traveler from D?

5Rectangle · 410 m

How far from A is the traveler after reducing complete laps?

6Pentagon · 365 m

A regular pentagon has side 40 m. How far past E is the traveler?

7Square meeting

A and B start at opposite vertices of an 80-m-side square, moving toward each other at 50 and 30 m/min. Meeting time?

8Square visibility

Earliest time the 90 and 70 m/min walkers share a side?

Mission 10

Exit ticket

Complete all five. The reflection is optional and is not automatically graded.

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1. Side length of a 360-m equilateral triangle (m)

2. Remainder after 725 m on a 360-m route (m)

3. Mission 4 first-meeting time (min)

4. Mission 5 distance from C when traveler B first reaches A (m)

5. On a 100-m-side square, coordinate 275 is on:

Optional reflection — not automatically graded

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Polygon-Track Coordinate Navigator

You completed the lesson checkpoints for perimeter coordinates, complete laps, directed arcs, and polygonal tracks. Review any steps for which you needed solution help.

Lesson 7.5 completed

Teaching notes

The coordinate strips, general locator, rectangle model, additional examples, feedback, workshop, and assessments are instructional scaffolds created to make the lesson self-contained.

For the square-wall extension, the page makes the implied visibility model explicit: the opaque wall blocks sight unless both walkers are on the same side. The first such interval begins at 50/3 minutes.