Perimeter and Distance Problems on the Coordinate Plane | Grade 8 Math | LevelKite

Earlier in this topic, you found the distance between two points. Now you'll use that skill to measure whole figures and to solve problems about real places.

A polygon is a closed figure made of straight sides. Each corner, where two sides meet, is a vertex (plural: vertices). The perimeter of a polygon is the total length around it: imagine walking along every side once and ending where you started.

On the coordinate plane, the horizontal axis is the $x$ -axis and the vertical axis is the $y$ -axis. Lengths are measured in grid units: one grid unit is the distance from one integer to the next on either axis.

Key concept

Perimeter is a sum of distances

Find each side's length from its endpoints' coordinates, then add the lengths, going around the figure once. Classify each side by its coordinates, not the drawing.

Horizontal (same $y$ -coordinates) or vertical (same $x$ -coordinates): the length is the absolute value of the difference of the coordinates that change.
Slanted: join its endpoints with a horizontal leg and a vertical leg. These legs meet at a right angle, as the axes do, and the side is the hypotenuse $c$ , opposite that angle. So $a^2 + b^2 = c^2$ for legs $a$ and $b$ ; take the nonnegative square root, since lengths are never negative.

The formula gives the perimeter $P$ of a four-sided polygon $ABCD$ , where $AB$ is the length from $A$ to $B$ . Keep all sides in one unit.

P = AB + BC + CD + DA

Triangle

ABC

with vertices

A(1, 1)

,

B(5, 1)

and

C(5, 4)

, the values used in the example.

Worked example

Perimeter of a triangle

Find the perimeter of triangle $ABC$ in the figure, which has vertices $A(1, 1)$ , $B(5, 1)$ and $C(5, 4)$ .

Find AB. Points A and B have the same y-coordinate, 1, so AB is horizontal. Only the x-coordinate changes, so the length is the absolute value of the difference of the x-coordinates.
$AB = |5 - 1| = 4$
Find BC. Points B and C have the same x-coordinate, 5, so BC is vertical. Only the y-coordinate changes, so use the y-coordinates the same way.
$BC = |4 - 1| = 3$
Find CA. AB is horizontal and BC is vertical, so they meet at a right angle at B. CA is opposite that angle, so it is the hypotenuse, and the Pythagorean theorem applies with legs 4 and 3. CA squared is 25, and both 5 and -5 square to 25, but CA is a length, so take the nonnegative square root.
$CA^2 = 4^2 + 3^2 = 16 + 9 = 25, \qquad CA = \sqrt{25} = 5$
Add the three side lengths to get the perimeter P. Going once around the triangle counts every side exactly once.
$P = AB + BC + CA = 4 + 3 + 5 = 12$

Answer

The perimeter of triangle $ABC$ is $12$ units.

Quadrilateral

ABCD

with

A(-2, -1)

,

B(1, 3)

,

C(7, 3)

and

D(2, -1)

. The helper right triangles for

AB

and

CD

have their right angles at

(-2, 3)

and

(7, -1)

.

Worked example

Perimeter with an exact and a rounded answer

Find the perimeter of quadrilateral $ABCD$ (a four-sided polygon) in the figure, with vertices $A(-2, -1)$ , $B(1, 3)$ , $C(7, 3)$ and $D(2, -1)$ . Give the exact perimeter, then round it to the nearest tenth of a unit.

Find the horizontal sides. B and C share the y-coordinate 3, and D and A share the y-coordinate -1, so BC and DA are horizontal. Take the absolute value of the difference of the x-coordinates, which keeps each length nonnegative.
$BC = |7 - 1| = 6, \qquad DA = |-2 - 2| = |-4| = 4$
Find AB with a helper right triangle that has its corner at (-2, 3). The leg from A to (-2, 3) is vertical, with length |3 - (-1)| = 4, and the leg from (-2, 3) to B is horizontal, with length |1 - (-2)| = 3, so the legs meet at a right angle. AB is opposite that angle, so it is the hypotenuse.
$AB^2 = 4^2 + 3^2 = 16 + 9 = 25, \qquad AB = \sqrt{25} = 5$
Find CD the same way, with the corner at (7, -1). The vertical leg from C to (7, -1) has length |-1 - 3| = 4, and the horizontal leg from (7, -1) to D has length |2 - 7| = 5. CD is opposite the right angle, so it is the hypotenuse. Since 41 is between the perfect squares 36 and 49, its square root is not a whole number, so keep it exact.
$CD^2 = 4^2 + 5^2 = 16 + 25 = 41, \qquad CD = \sqrt{41}$
Add the four sides in order around the figure. DA closes the path back to A, so each side is counted exactly once. This sum is the exact perimeter.
$P = AB + BC + CD + DA = 5 + 6 + \sqrt{41} + 4 = 15 + \sqrt{41}$
Now locate the square root of 41. For positive numbers, a larger number has a larger square, so you can compare squares. 41 is between the squares of 6.4 and 6.5, so the root is between 6.4 and 6.5. To round, compare 41 with the square of the midpoint, 6.45, which is halfway between them. 41 is smaller, so the root is less than 6.45.
$6.4^2 = 40.96 < 41 < 41.6025 = 6.45^2 < 42.25 = 6.5^2$
So the root is between 6.4 and 6.45. Adding 15 to every part keeps the order, so the perimeter is between 21.4 and 21.45, which rounds to 21.4. The symbol ≈ means 'is approximately equal to'.
$6.4 < \sqrt{41} < 6.45 \;\Rightarrow\; 21.4 < 15 + \sqrt{41} < 21.45, \qquad P \approx 21.4$

Answer

The exact perimeter of $ABCD$ is $15 + \sqrt{41}$ units, which is about $21.4$ units to the nearest tenth.

In many word problems, the coordinate plane is a map. Before you calculate, decide three things:

Directions: what each axis means. On the map below, the positive $x$ -direction is east and the positive $y$ -direction is north.
Scale: the real distance that one grid unit stands for.
Which length: a straight-line distance is the length of the segment joining two points. A trip along streets that run only east-west or north-south follows horizontal and vertical segments, so its length is the sum of those segments' lengths.

Find lengths in grid units first, then multiply by the scale. This works when one grid unit stands for the same real distance in both directions, because then every length on the map is multiplied by the same number. Add or compare lengths only when they are in the same unit.

The store is Point A at

(-4, -3)

and the home is Point B at

(2, 5)

. The truck's route is the horizontal and vertical sides through

(2, -3)

; the drone's path is the slanted side.

Worked example

Word problem: truck route or drone flight

On the town map in the figure, the positive $x$ -direction is east, the positive $y$ -direction is north, and one grid unit stands for $100$ meters in both directions. A store is at Point A $(-4, -3)$ and a home is at Point B $(2, 5)$ . A delivery truck drives east along a street to the corner at $(2, -3)$ , then north along another street to the home. A drone flies straight from the store to the home. How far does each travel, and how much shorter is the drone's trip?

Find the truck's route in grid units. The drive east keeps y = -3, so it is horizontal; the drive north keeps x = 2, so it is vertical. The truck travels both parts, so add their lengths.
$|2 - (-4)| + |5 - (-3)| = 6 + 8 = 14 \text{ units}$
Let d be the drone's distance in grid units. The two parts of the truck's route are a horizontal leg and a vertical leg that meet at a right angle at (2, -3). The drone's straight path is opposite that angle, so it is the hypotenuse; take the nonnegative square root.
$d = \sqrt{6^2 + 8^2} = \sqrt{36 + 64} = \sqrt{100} = 10 \text{ units}$
Convert to meters. One grid unit stands for 100 meters in both directions, so multiply each length by 100.
$14 \times 100 = 1{,}400 \text{ m}, \qquad 10 \times 100 = 1{,}000 \text{ m}$
Both lengths are now in meters, so subtract to compare them.
$1{,}400 - 1{,}000 = 400 \text{ m}$

Answer

The truck travels 1,400 meters and the drone flies 1,000 meters, so the drone's trip is 400 meters shorter.

Takeaway: A perimeter on the coordinate plane is a sum of distances. Find each side from its endpoints' coordinates. For a horizontal or vertical side, take the absolute value of the difference of the coordinates that change. For a slanted side, use $a^2 + b^2 = c^2$ on a right triangle with a horizontal leg and a vertical leg, where the side is the hypotenuse, opposite the right angle; take the nonnegative square root. Add every side once, and keep exact values until you round. In a word problem, decide which length is asked for, work in grid units, then convert with the scale.

Next, you'll verify whether a triangle on the coordinate plane is a right triangle.