Graph Shapes and Fix Mistakes on the Coordinate Plane | Grade 5 Math | LevelKite

You already know how to plot one point from an ordered pair such as $(5, 1)$ . Now you will plot a few points and join them to make a shape.

A corner of a shape is called a vertex. The word for more than one vertex is vertices. A triangle has 3 sides and 3 vertices. A quadrilateral is a shape with 4 straight sides, so it has 4 vertices.

Once you know where the vertices are, you know the whole shape. Each side is a straight line from one vertex to the next one in the list. The last side goes from the last vertex back to the first.

In every figure here, the $x$ -axis runs along the bottom and the $y$ -axis runs up the left side. Each grid square is 1 unit wide and 1 unit tall.

Key concept

The ordered pairs make the shape

The ordered pairs of the vertices tell you exactly where a shape is on the coordinate plane.

Plot each vertex from the origin, the point $(0, 0)$ where the two axes meet. Move right by the $x$ -coordinate, the first number. Then move up by the $y$ -coordinate, the second number. Join the vertices in the order they are listed, and join the last one back to the first.

The same ordered pairs also tell you how long a side is when it runs straight across or straight up and down. And they let you check any point that looks wrong.

Worked example

Graph a triangle

Graph the triangle with vertices $A(2, 1)$ , $B(6, 1)$ and $C(6, 4)$ .

Plot vertex A. Start at the origin. The x-coordinate is 2, so move 2 units right. The y-coordinate is 1, so move 1 unit up. Mark Point A there.
$A(2, 1)$
Plot vertex B. Every point is counted from the origin, so go back there first. The x-coordinate is 6, so move 6 units right. The y-coordinate is 1, so move 1 unit up. Mark Point B.
$B(6, 1)$
Plot vertex C the same way. From the origin, move 6 units right because the x-coordinate is 6. Then move 4 units up because the y-coordinate is 4. Mark Point C.
$C(6, 4)$
Join the vertices in the order they are listed, because each side connects two vertices that are next to each other in the list. Draw a straight line from A to B, then from B to C. Then join C back to A to close the triangle.
$A \rightarrow B \rightarrow C \rightarrow A$

Answer

The triangle has its vertices at $(2, 1)$ , $(6, 1)$ and $(6, 4)$ . Its three sides are $AB$ , $BC$ and $CA$ .

Point A is at

(2, 1)

, Point B is at

(6, 1)

and Point C is at

(6, 4)

. The sides join A to B, B to C and C back to A.

Now use the coordinates to measure the triangle.

A horizontal side runs straight across, left to right. A vertical side runs straight up and down.

Look at side $AB$ in the figure above. It goes from $(2, 1)$ to $(6, 1)$ . Both $y$ -coordinates are 1, so Point A and Point B are at the same height. That makes the side horizontal. Point A is 2 units to the right of the $y$ -axis. Point B is 6 units to the right of it. The gap between them is the length of the side: $6 - 2 = 4$ units. Count the squares along $AB$ to check. There are 4.

Side $BC$ goes from $(6, 1)$ to $(6, 4)$ . Both $x$ -coordinates are 6, so Point C is straight above Point B. That makes the side vertical. Point B is 1 unit above the $x$ -axis and Point C is 4 units above it, so the length is $4 - 1 = 3$ units.

Worked example

Graph a quadrilateral and measure its sides

Graph the quadrilateral with vertices $D(1, 2)$ , $E(7, 2)$ , $F(7, 6)$ and $G(4, 6)$ . Then find the lengths of sides $DE$ , $EF$ and $FG$ .

Plot each vertex from the origin. The x-coordinate tells you how far to move right and the y-coordinate tells you how far to move up. For example, G(4, 6) is 4 units right and 6 units up.
$D(1, 2) \quad E(7, 2) \quad F(7, 6) \quad G(4, 6)$
Join the vertices in the listed order, then join G back to D to close the shape. The order matters. A line from D straight to F would cut across the inside of the shape, so it is not a side.
$D \rightarrow E \rightarrow F \rightarrow G \rightarrow D$
Side DE goes from (1, 2) to (7, 2). The y-coordinates are both 2, so the side is horizontal. Subtract the x-coordinates to find the gap between its two ends.
$7 - 1 = 6$
Side EF goes from (7, 2) to (7, 6). The x-coordinates are both 7, so the side is vertical. Subtract the y-coordinates.
$6 - 2 = 4$
Side FG goes from (7, 6) to (4, 6). The y-coordinates are both 6, so the side is horizontal. Take the smaller x-coordinate from the larger one.
$7 - 4 = 3$

Answer

Quadrilateral $DEFG$ is graphed by joining its four vertices in order. Side $DE$ is $6$ units long, side $EF$ is $4$ units long and side $FG$ is $3$ units long.

Point D is at

(1, 2)

, Point E is at

(7, 2)

, Point F is at

(7, 6)

and Point G is at

(4, 6)

. Count the squares along each side:

DE

has 6,

EF

has 4 and

FG

has 3. Side

GD

is slanted.

A point can look neat and still be in the wrong place. So check your work by reading each point back.

Start at the origin. Count how many units right the point is. Then count how many units up. Write the two counts as an ordered pair, right first and up second. It should match the ordered pair you were given.

If it does not match, look at how it is different. That shows you what went wrong. Two mistakes happen a lot: swapping the two coordinates, and starting the count at 1 instead of at the origin.

Worked example

Find and fix a mistake

Mia has to graph the triangle with vertices $(2, 1)$ , $(5, 1)$ and $(3, 6)$ . She plots points at $(2, 1)$ , $(5, 1)$ and $(6, 3)$ . Find her mistake and correct it.

Compare each point Mia plotted with the vertex it should be. Her first point, (2, 1), matches. Her second point, (5, 1), matches. Her third point is (6, 3), but the vertex should be (3, 6). The mistake is in the third point.
$\text{plotted: } (6, 3) \qquad \text{given: } (3, 6)$
Look at how the two pairs differ. They use the same numbers, 3 and 6, in the opposite order. So Mia swapped the coordinates: she moved 6 right and 3 up. That fails because the first number is always the x-coordinate, the move to the right. The pair (3, 6) means 3 right and 6 up, which is a different place.
Correct the point. Start at the origin. Move 3 units right for the x-coordinate, then 6 units up for the y-coordinate. Mark the vertex there and join it to the other two vertices.
$3 \text{ right, } 6 \text{ up} \rightarrow (3, 6)$

Answer

Mia swapped the $x$ - and $y$ -coordinates of the third vertex. It belongs at $(3, 6)$ , not at $(6, 3)$ .

The triangle is drawn with its third vertex in the correct place,

(3, 6)

. The lone point at

(6, 3)

is where Mia plotted that vertex.

Takeaway. The ordered pairs of the vertices hold the whole shape. Plot each vertex from the origin, right by the $x$ -coordinate and up by the $y$ -coordinate. Then join the vertices in order. If two vertices have the same $y$ -coordinate, the side between them is horizontal, so subtract the $x$ -coordinates to find its length. If they have the same $x$ -coordinate, the side is vertical, so subtract the $y$ -coordinates. When a point looks wrong, read it back from the origin and compare it with its ordered pair.