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Distances, routes and shapes on the plane

الأسبوع 25 · المسافات على المستوى الإحداثي ورسم الأشكال من نقاط · 3 دروس
2

Distance on the coordinate plane: routes and maps

Discover Mathematics P6: Term 2, Unit 11, Lessons 4–6
GoalUse distances along grid lines to follow routes and find places on a grid map.
New words
route · the path from one place to another along grid linesgrid map · a map drawn on a coordinate planescale · how much real distance one unit on the map stands for
Explanation

A grid map is a coordinate plane in which each place is a point. Roads go along the grid lines, so a route between two places is made of horizontal and vertical pieces. To find the length of a route, find the distance of each piece (same x or same y) and add them. The scale tells what one unit means: if 1 unit is 100 m, then 12 units are 1,200 m. We can also look for places at a given distance: from (1, 2), the points 4 units away along the horizontal line are (5, 2) and (−3, 2), one on each side. We never need a distance formula; we only count along the axes.

★Remember
A route = horizontal pieces + vertical pieces. Find each piece with absolute values, add them, then use the scale.
Worked examples
xy−6−5−4−3−2−1123456−6−5−4−3−2−11234560PLB
On a map 1 unit = 100 m. The post office P is at (−4, 3), the library L at (−4, −2) and the bakery B at (3, −2). P to L: same x, 3 + 2 = 5 units. L to B: same y, 4 + 3 = 7 units. The route P → L → B is 5 + 7 = 12 units = 1,200 m.
A school S is at (1, 2). Which points on the same horizontal line are 4 units from S? Go 4 right: (5, 2). Go 4 left: (−3, 2). There are two answers.
A walker goes from (−3, −3) to (4, −3) and then to (4, 2). First piece: 3 + 4 = 7 units. Second piece: 3 + 2 = 5 units. Total 12 units. If 1 unit is 50 m, the walk is 12 × 50 = 600 m.
Class activity

Draw my town. On squared paper draw axes and place five places (home, school, shop, park, club) at points with whole-number coordinates. Choose a scale. Write the length of the route from home to school along the grid lines.

Practice
1
A house is at (2, 3) and a shop at (2, −4). How far apart are they?
2
A park is at (−5, −1) and a pool at (−5, −6). Find the distance.
3
Find the points 3 units from (−2, −1) on the vertical line through it.
4
Look at the map. How long is the route from P to L to B in units? If 1 unit is 100 m, how many metres?
xy−6−5−4−3−2−1123456−6−5−4−3−2−11234560PLB
5
Salma walks from (1, 1) to (1, 6) and then to (−2, 6). How many units does she walk?