The Coordinate Plane

Structure of the Four-Quadrant Coordinate Plane

The coordinate plane is a flat grid that helps us show the exact location of points. It combines ideas from number lines and geometry.

In 6th grade, you will use the coordinate plane to plot points, name points, and understand where points are located in different parts of the grid. A four-quadrant coordinate plane includes both positive and negative numbers.

1. The two number lines that make the coordinate plane

The coordinate plane is made from two number lines that cross each other:

• The x-axis is the horizontal number line.
• The y-axis is the vertical number line.

These two axes meet at one special point called the origin.

The origin is written as \((0,0)\). It means the point is 0 units along the x-axis and 0 units along the y-axis.

2. Positive and negative directions

Each axis has a positive direction and a negative direction:

• On the x-axis, numbers to the right are positive, and numbers to the left are negative.
• On the y-axis, numbers up are positive, and numbers down are negative.

This means:

• Right = positive x
• Left = negative x
• Up = positive y
• Down = negative y

3. Ordered pairs

A point on the coordinate plane is named by an ordered pair. An ordered pair looks like \((x,y)\).

The first number tells how far to move along the x-axis. The second number tells how far to move along the y-axis.

Always remember: x first, then y.

You can think of it like this:

1. Start at the origin, \((0,0)\).
2. Move left or right using the x-value.
3. Then move up or down using the y-value.

For example, to plot \((3,2)\):

• Start at \((0,0)\).
• Move 3 units right.
• Move 2 units up.

4. The four quadrants

The x-axis and y-axis divide the plane into four sections called quadrants.

The quadrants are labeled using Roman numerals:

• Quadrant I: top right
• Quadrant II: top left
• Quadrant III: bottom left
• Quadrant IV: bottom right

They are numbered in a counterclockwise direction, starting from the top right.

The signs of the coordinates in each quadrant are:

• Quadrant I: \((+,+)\)
• Quadrant II: \((-,+)\)
• Quadrant III: \((-,-)\)
• Quadrant IV: \((+,-)\)

This is a very helpful pattern to memorize.

Here is the pattern in a simple layout:

$$ \begin{array}{c|c} \text{Quadrant II }(-,+) & \text{Quadrant I }(+,+) \\ \hline \text{Quadrant III }(-,-) & \text{Quadrant IV }(+,-) \end{array} $$

5. Points on an axis are not in a quadrant

If a point lies exactly on the x-axis or y-axis, it is not in any quadrant.

Why? Because quadrants are the four regions between the axes, not on the axes.

Examples:

• \((4,0)\) is on the x-axis because the y-coordinate is 0.
• \((0,-3)\) is on the y-axis because the x-coordinate is 0.
• \((0,0)\) is the origin.

6. How to tell where a point is located

To describe a point, ask these questions:

1. Is the x-value positive, negative, or 0?
2. Is the y-value positive, negative, or 0?
3. Use the signs to decide the quadrant or axis.

For example:

• If x is negative and y is positive, the point is in Quadrant II.
• If x is positive and y is negative, the point is in Quadrant IV.
• If y is 0, the point is on the x-axis.

Worked Example 1: Plotting a point in Quadrant I

Plot the point \((4,3)\).

Step 1: Start at the origin, \((0,0)\).

Step 2: The x-value is 4, so move 4 units to the right.

Step 3: The y-value is 3, so move 3 units up.

The point is in Quadrant I because both coordinates are positive.

Worked Example 2: Naming the quadrant of a point

In which quadrant is the point \((-5,2)\)?

The x-coordinate is negative, so the point is left of the y-axis.

The y-coordinate is positive, so the point is above the x-axis.

A point that is left and up is in Quadrant II.

Answer: \((-5,2)\) is in Quadrant II.

Worked Example 3: Point on an axis

Where is the point \((0,-6)\) located?

The x-coordinate is 0, so the point does not move left or right. That means it lies on the y-axis.

The y-coordinate is \(-6\), so it is 6 units below the origin.

Because it is on an axis, it is not in any quadrant.

Answer: \((0,-6)\) is on the y-axis.

Worked Example 4: Comparing different points

Tell where each point is located: \((2,-4)\), \((-3,-1)\), and \((5,0)\).

Point 1: \((2,-4)\)

• x is positive
• y is negative

So the point is in Quadrant IV.

Point 2: \((-3,-1)\)

• x is negative
• y is negative

So the point is in Quadrant III.

Point 3: \((5,0)\)

• y is 0

So the point is on the x-axis, not in a quadrant.

7. Common mistakes to avoid

• Mixing up x and y: Remember, x comes first and y comes second.
• Moving in the wrong direction: Negative x means left, and negative y means down.
• Putting axis points in quadrants: If x = 0 or y = 0, the point is on an axis.
• Starting somewhere other than the origin: Always begin at \((0,0)\) when plotting.

8. Quick guide for quadrants

• Quadrant I: right and up \(\rightarrow (+,+)\)
• Quadrant II: left and up \(\rightarrow (-,+)\)
• Quadrant III: left and down \(\rightarrow (-,-)\)
• Quadrant IV: right and down \(\rightarrow (+,-)\)

A simple way to remember is:

• Start in the top right with all positive.
• Move counterclockwise around the plane.

Summary

The four-quadrant coordinate plane is made by the horizontal x-axis and the vertical y-axis. These axes meet at the origin, \((0,0)\), and divide the plane into four quadrants.

Every point is written as an ordered pair, \((x,y)\). The x-value tells left or right movement, and the y-value tells up or down movement.

By looking at whether each coordinate is positive, negative, or 0, you can tell whether a point is in Quadrant I, II, III, IV, on an axis, or at the origin.

Plotting Ordered Pairs in All Quadrants

Have you ever used a map with a grid to find an exact location? The coordinate plane works in a similar way. It helps us show exactly where a point is by using a pair of numbers called an ordered pair.

In this lesson, you will learn how to read and plot ordered pairs on the coordinate plane in all four quadrants. By the end, you should be able to find points like \((3, 2)\), \((-4, 5)\), \((-2, -3)\), and \((6, -1)\).

1. What is the coordinate plane?

The coordinate plane is a flat grid made by two number lines that cross.

• The x-axis is the horizontal number line.
• The y-axis is the vertical number line.
• They cross at \((0, 0)\), which is called the origin.

The coordinate plane is divided into 4 sections called quadrants.

1. Quadrant I: top right
2. Quadrant II: top left
3. Quadrant III: bottom left
4. Quadrant IV: bottom right

You can remember the order by starting in the top right and moving counterclockwise: I, II, III, IV.

2. What is an ordered pair?

An ordered pair is written like this:

$$ (x, y) $$

The first number tells how far to move left or right on the x-axis. The second number tells how far to move up or down on the y-axis.

• If the x-value is positive, move right.
• If the x-value is negative, move left.
• If the y-value is positive, move up.
• If the y-value is negative, move down.

Important: The order matters. \((2, 5)\) is not the same point as \((5, 2)\).

3. How to plot an ordered pair

To plot a point \((x, y)\), follow these steps:

1. Start at the origin, \((0, 0)\).
2. Look at the x-value first and move left or right.
3. Then look at the y-value and move up or down.
4. Place a dot where you land.

Always remember: x first, then y.

4. Signs in each quadrant

The signs of the coordinates help you know which quadrant a point is in.

• Quadrant I: \((+, +)\)
• Quadrant II: \((-, +)\)
• Quadrant III: \((-, -)\)
• Quadrant IV: \((+, -)\)

For example:

• \((4, 3)\) is in Quadrant I because both numbers are positive.
• \((-4, 3)\) is in Quadrant II because x is negative and y is positive.
• \((-4, -3)\) is in Quadrant III because both numbers are negative.
• \((4, -3)\) is in Quadrant IV because x is positive and y is negative.

5. Points on the axes

Not every point is in a quadrant. Some points lie on an axis.

• If the y-value is 0, the point is on the x-axis.
• If the x-value is 0, the point is on the y-axis.
• The point \((0, 0)\) is the origin.

Examples:

• \((5, 0)\) is on the x-axis.
• \((0, -2)\) is on the y-axis.
• \((0, 0)\) is the origin.

Worked Example 1: Plot a point in Quadrant I

Plot the point \((3, 4)\).

Step 1: Start at \((0, 0)\).

Step 2: The x-value is 3, so move 3 units to the right.

Step 3: The y-value is 4, so move 4 units up.

Step 4: Put a dot there.

The point \((3, 4)\) is in Quadrant I.

Worked Example 2: Plot a point in Quadrant II

Plot the point \((-2, 5)\).

Step 1: Start at the origin.

Step 2: The x-value is \(-2\), so move 2 units to the left.

Step 3: The y-value is 5, so move 5 units up.

Step 4: Put a dot there.

The point \((-2, 5)\) is in Quadrant II.

Worked Example 3: Plot a point in Quadrant III

Plot the point \((-4, -3)\).

Step 1: Start at \((0, 0)\).

Step 2: The x-value is \(-4\), so move 4 units left.

Step 3: The y-value is \(-3\), so move 3 units down.

Step 4: Put a dot there.

The point \((-4, -3)\) is in Quadrant III.

Worked Example 4: Plot a point in Quadrant IV and identify it

Plot the point \((6, -2)\).

Step 1: Start at the origin.

Step 2: The x-value is 6, so move 6 units right.

Step 3: The y-value is \(-2\), so move 2 units down.

Step 4: Put a dot there.

The point \((6, -2)\) is in Quadrant IV.

6. How to name a point from a graph

Sometimes a point is already plotted on the graph, and you need to write its ordered pair.

To do this:

1. Start at the origin.
2. Count how far the point is left or right. That is the x-value.
3. Then count how far the point is up or down. That is the y-value.
4. Write the point as \((x, y)\).

For example, if a point is 3 units left and 2 units down from the origin, its coordinates are:

$$ (-3, -2) $$

7. Common mistakes to avoid

• Switching the numbers: Remember, x comes first and y comes second.
• Starting somewhere other than the origin: Always begin at \((0, 0)\).
• Moving the wrong direction: Positive x means right, negative x means left, positive y means up, and negative y means down.
• Forgetting about signs: A negative sign changes the direction.

8. Quick check with mental practice

Try to decide where each point goes before plotting it.

• \((2, 3)\): right 2, up 3, so it is in Quadrant I
• \((-5, 1)\): left 5, up 1, so it is in Quadrant II
• \((-1, -4)\): left 1, down 4, so it is in Quadrant III
• \((4, -6)\): right 4, down 6, so it is in Quadrant IV
• \((0, 7)\): on the y-axis
• \((-3, 0)\): on the x-axis

9. Helpful memory tips

• x first, then y
• Across first, then up or down
• Start at the origin every time
• Use the signs to find the quadrant

Summary

An ordered pair tells the exact location of a point on the coordinate plane. The first number, x, tells left or right, and the second number, y, tells up or down. By starting at the origin and following the ordered pair carefully, you can plot points in any of the four quadrants or on an axis.

When you see a point, check the signs of the coordinates to help identify its quadrant. With practice, plotting ordered pairs becomes a quick and useful skill in both algebra and geometry.

Reflecting Points Across Axes

In this lesson, you will learn how to reflect points on the coordinate plane across the x-axis and the y-axis.

A reflection is like looking at a shape or point in a mirror. The reflected point ends up the same distance from the axis as the original point, but on the opposite side.

To understand reflections, we first need to remember how the coordinate plane works. A point is written as an ordered pair \,\((x, y)\). The first number tells how far to move left or right, and the second number tells how far to move up or down.

For example, the point \,\((3, 2)\) means move 3 units to the right and 2 units up. The point \,\((-4, 1)\) means move 4 units to the left and 1 unit up.

The x-axis is the horizontal axis. It goes left and right. Every point on the x-axis has a y-coordinate of 0.

The y-axis is the vertical axis. It goes up and down. Every point on the y-axis has an x-coordinate of 0.

When you reflect a point, one coordinate stays the same and the other coordinate changes sign.

Reflection across the x-axis: the x-coordinate stays the same, and the y-coordinate changes to its opposite.

Rule:

$$ (x, y) \rightarrow (x, -y) $$

This means a point above the x-axis moves the same distance below it, or a point below the x-axis moves the same distance above it.

Reflection across the y-axis: the y-coordinate stays the same, and the x-coordinate changes to its opposite.

Rule:

$$ (x, y) \rightarrow (-x, y) $$

This means a point to the right of the y-axis moves the same distance to the left, or a point to the left of the y-axis moves the same distance to the right.

Here is a helpful way to remember:

• x-axis: change the y
• y-axis: change the x

You can also think about it this way: the axis you reflect across acts like a mirror. The point keeps the same distance from that mirror line.

Worked Example 1

Reflect the point \,\((4, 3)\) across the x-axis.

Use the rule for the x-axis:

$$ (x, y) \rightarrow (x, -y) $$

The x-coordinate stays 4. The y-coordinate changes from 3 to \,\(-3\).

So the reflected point is:

$$ (4, 3) \rightarrow (4, -3) $$

Worked Example 2

Reflect the point \,\((-2, 5)\) across the y-axis.

Use the rule for the y-axis:

$$ (x, y) \rightarrow (-x, y) $$

The x-coordinate changes from \,\(-2\) to 2. The y-coordinate stays 5.

So the reflected point is:

$$ (-2, 5) \rightarrow (2, 5) $$

Worked Example 3

Reflect the point \,\((-6, -1)\) across the x-axis.

The x-coordinate stays the same. The y-coordinate changes sign.

$$ (-6, -1) \rightarrow (-6, 1) $$

Even though the y-coordinate is already negative, it still changes to its opposite.

Worked Example 4

A triangle has a vertex at \,\((1, 4)\), another at \,\((3, 2)\), and another at \,\((2, 5)\). Reflect the triangle across the y-axis.

Reflect each point one at a time using \,\((x, y) \rightarrow (-x, y)\).

• \((1, 4) \rightarrow (-1, 4)\)
• \((3, 2) \rightarrow (-3, 2)\)
• \((2, 5) \rightarrow (-2, 5)\)

So the reflected triangle has vertices at:

$$ (-1, 4), \ (-3, 2), \ (-2, 5) $$

This shows that to reflect a whole figure, you reflect every point in the figure.

Points on an Axis

If a point is already on the axis you are reflecting across, it does not move.

• If a point is on the x-axis, then \,\(y = 0\).
• If a point is on the y-axis, then \,\(x = 0\).

For example, reflect \,\((5, 0)\) across the x-axis.

Using the rule gives:

$$ (5, 0) \rightarrow (5, 0) $$

The point stays in the same place because it is already on the x-axis.

Common Mistakes to Avoid

• Do not change both coordinates when reflecting across just one axis.
• Across the x-axis, only the y-value changes.
• Across the y-axis, only the x-value changes.
• Be careful with negative numbers. Changing sign means positive becomes negative, and negative becomes positive.

Quick Check

• Reflect \,\((2, -7)\) across the x-axis: \,\((2, 7)\)
• Reflect \,\((2, -7)\) across the y-axis: \,\((-2, -7)\)
• Reflect \,\((-4, 0)\) across the x-axis: \,\((-4, 0)\)

Summary

Reflecting a point means flipping it across an axis like a mirror image.

• Across the x-axis: $$ (x, y) \rightarrow (x, -y) $$
• Across the y-axis: $$ (x, y) \rightarrow (-x, y) $$

Remember: x-axis changes y, and y-axis changes x. If you reflect a figure, reflect each point in the figure using the same rule.

Calculating Distance on the Coordinate Plane

The coordinate plane is a grid made from two number lines. One number line goes left and right. This is the x-axis. The other number line goes up and down. This is the y-axis.

A point on the coordinate plane is written as an ordered pair, like \((3, 5)\). The first number tells how far to move left or right on the x-axis. The second number tells how far to move up or down on the y-axis.

In this lesson, you will learn how to find the distance between two points when they are lined up horizontally or vertically. This means the points share the same x-coordinate or the same y-coordinate.

Main Idea: If two points are on the same horizontal or vertical line, you can find the distance by subtracting the coordinates that are different and taking the absolute value.

Absolute value means the distance from 0 on a number line, so it is always positive or 0. For example:

• \(|5| = 5\)
• \(|-5| = 5\)
• \(|3 - 8| = |-5| = 5\)

When the points are horizontal:

• The y-coordinates are the same.
• You find the distance by comparing the x-coordinates.
• Use: $$\text{distance} = |x_2 - x_1|$$

When the points are vertical:

• The x-coordinates are the same.
• You find the distance by comparing the y-coordinates.
• Use: $$\text{distance} = |y_2 - y_1|$$

You can think of this as counting how many units apart the points are on a straight line. If the points go left and right, compare the x-values. If the points go up and down, compare the y-values.

How to decide what to do:

1. Look at the two points.
2. Check whether the x-values are the same or the y-values are the same.
3. If the y-values are the same, subtract the x-values.
4. If the x-values are the same, subtract the y-values.
5. Take the absolute value so the answer is positive.

Worked Example 1: Horizontal Distance

Find the distance between \((2, 4)\) and \((7, 4)\).

Step 1: Look at the coordinates. Both points have the same y-coordinate: \(4\).

That means the points are on a horizontal line.

Step 2: Subtract the x-coordinates and use absolute value.

$$|7 - 2| = |5| = 5$$

Answer: The distance is 5 units.

Worked Example 2: Vertical Distance

Find the distance between \((-3, 1)\) and \((-3, 6)\).

Step 1: Both points have the same x-coordinate: \(-3\).

That means the points are on a vertical line.

Step 2: Subtract the y-coordinates and use absolute value.

$$|6 - 1| = |5| = 5$$

Answer: The distance is 5 units.

Worked Example 3: Using Negative Numbers

Find the distance between \((-4, -2)\) and \((3, -2)\).

Step 1: The y-coordinates are the same: \(-2\).

So the points are horizontal.

Step 2: Subtract the x-coordinates.

$$|3 - (-4)| = |3 + 4| = |7| = 7$$

Answer: The distance is 7 units.

This example shows why absolute value is helpful. Even when coordinates are negative, distance is still a positive number.

Worked Example 4: Vertical Distance with Negative Numbers

Find the distance between \((5, -3)\) and \((5, 4)\).

Step 1: The x-coordinates are the same: \(5\).

So the points are vertical.

Step 2: Subtract the y-coordinates.

$$|4 - (-3)| = |4 + 3| = |7| = 7$$

Answer: The distance is 7 units.

Important Tips

• Distance is always given in units.
• If the points are horizontal, compare the x-values.
• If the points are vertical, compare the y-values.
• Use absolute value so your answer is not negative.
• You can subtract in either order because absolute value makes the result positive.

Common Mistakes to Avoid

• Mistake 1: Subtracting the wrong coordinates. Always check which coordinates are the same first.
• Mistake 2: Forgetting absolute value. Distance cannot be negative.
• Mistake 3: Mixing up horizontal and vertical lines.

For example, if you want the distance between \((1, 8)\) and \((1, 2)\), do not subtract the x-coordinates. The x-coordinates are the same, so you must subtract the y-coordinates:

$$|8 - 2| = 6$$

Quick Check

• Distance between \((0, 3)\) and \((6, 3)\): $$|6 - 0| = 6$$
• Distance between \((-2, -5)\) and \((-2, 1)\): $$|1 - (-5)| = 6$$

Summary

To find the distance between two points on the coordinate plane, first see whether the points are horizontal or vertical. If the points have the same y-coordinate, find the absolute difference of the x-coordinates. If the points have the same x-coordinate, find the absolute difference of the y-coordinates.

In short:

• Same y-coordinate: $$\text{distance} = |x_2 - x_1|$$
• Same x-coordinate: $$\text{distance} = |y_2 - y_1|$$

With practice, you will get faster at spotting whether to compare x-values or y-values. Remember: distance is how far apart points are, so the answer should always be positive.

Drawing and Analyzing Polygons on the Grid

In this lesson, you will learn how to plot points on a coordinate grid, connect those points to make polygons, and use the coordinates to study the shape.

A polygon is a closed shape made of straight sides. Examples include triangles, rectangles, squares, and pentagons.

When a polygon is drawn on a coordinate plane, each corner of the shape is called a vertex. The plural of vertex is vertices.

By looking at the coordinates of the vertices, you can:

• draw the polygon correctly,
• find missing vertices,
• find side lengths when sides are horizontal or vertical,
• and calculate the perimeter.

1. Review of the Coordinate Plane

A coordinate plane is a grid made by two number lines:

• the x-axis, which goes left and right,
• the y-axis, which goes up and down.

The point where the axes cross is called the origin. Its coordinates are \((0,0)\).

Every point is named with an ordered pair \((x,y)\):

• The x-coordinate tells how far to move left or right.
• The y-coordinate tells how far to move up or down.

To plot \((3,2)\), start at the origin, move 3 units right, then 2 units up.

To plot \((-4,1)\), start at the origin, move 4 units left, then 1 unit up.

2. How to Draw a Polygon on the Grid

To draw a polygon from coordinates, follow these steps:

1. Plot each given point carefully.
2. Label the points with letters if needed.
3. Connect the points in the order given.
4. Make sure the last point connects back to the first point so the shape is closed.

If the points are not connected in the correct order, you may get the wrong shape.

3. Finding Side Lengths on the Grid

In 6th Grade, the easiest side lengths to find are horizontal and vertical sides.

A horizontal side goes left to right. Its endpoints have the same y-coordinate.

To find the length of a horizontal side, subtract the x-coordinates:

$$\text{length} = |x_2 - x_1|$$

A vertical side goes up and down. Its endpoints have the same x-coordinate.

To find the length of a vertical side, subtract the y-coordinates:

$$\text{length} = |y_2 - y_1|$$

The absolute value bars mean the distance is always positive.

For example:

• From \((2,5)\) to \((7,5)\), the side is horizontal, and the length is \(|7-2|=5\).
• From \((4,1)\) to \((4,6)\), the side is vertical, and the length is \(|6-1|=5\).

4. Finding Perimeter

The perimeter of a polygon is the total distance around it.

To find perimeter:

1. Find the length of each side.
2. Add the side lengths together.

If a rectangle has side lengths 6 and 4, then its perimeter is:

$$6+4+6+4=20$$

5. Finding Missing Vertices

Sometimes a problem gives some vertices of a polygon and asks for the missing one.

When the shape is a rectangle or square with sides parallel to the axes:

• points on the same vertical side share the same x-coordinate,
• points on the same horizontal side share the same y-coordinate.

This helps you figure out the missing point by matching x-values and y-values.

For example, if three vertices of a rectangle are \((1,2)\), \((1,6)\), and \((5,6)\), then the missing vertex must be \((5,2)\).

Why? The left side uses \(x=1\), the right side uses \(x=5\), the top uses \(y=6\), and the bottom uses \(y=2\).

Worked Example 1: Drawing a Triangle

Plot the points \(A(1,1)\), \(B(5,1)\), and \(C(3,4)\). Then connect them to make a polygon.

Step 1: Plot each point.

• \(A(1,1)\): right 1, up 1
• \(B(5,1)\): right 5, up 1
• \(C(3,4)\): right 3, up 4

Step 2: Connect \(A\) to \(B\), \(B\) to \(C\), and \(C\) back to \(A\).

Result: The polygon is a triangle.

Notice that \(A\) and \(B\) have the same y-coordinate, so side \(AB\) is horizontal.

The length of \(AB\) is:

$$|5-1|=4$$

Worked Example 2: Rectangle and Perimeter

Draw rectangle \(PQRS\) with vertices \(P(2,2)\), \(Q(8,2)\), \(R(8,5)\), and \(S(2,5)\). Find its perimeter.

Step 1: Plot the points and connect them in order.

Step 2: Find the side lengths.

Side \(PQ\) is horizontal because both points have \(y=2\).

$$PQ = |8-2| = 6$$

Side \(QR\) is vertical because both points have \(x=8\).

$$QR = |5-2| = 3$$

Opposite sides of a rectangle are equal, so:

• \(RS = 6\)
• \(SP = 3\)

Step 3: Add all side lengths.

$$6+3+6+3=18$$

Answer: The perimeter is 18 units.

Worked Example 3: Finding a Missing Vertex

Three vertices of a rectangle are \(A(4,1)\), \(B(4,6)\), and \(C(9,6)\). Find the fourth vertex.

Step 1: Look at the x-coordinates.

• \(A\) and \(B\) both have \(x=4\), so they form a vertical side.
• \(C\) has \(x=9\), so the opposite vertical side must use \(x=9\).

Step 2: Look at the y-coordinates.

• \(B\) and \(C\) both have \(y=6\), so they form a horizontal side.
• The bottom side must use the same y-coordinate as point \(A\), which is \(y=1\).

Step 3: Combine the needed coordinates.

The missing vertex is \((9,1)\).

Answer: The fourth vertex is \((9,1)\).

Worked Example 4: Drawing and Analyzing a Polygon

Plot the vertices \(M(1,3)\), \(N(6,3)\), \(O(6,7)\), and \(P(1,7)\). Then name the polygon and find its perimeter.

Step 1: Plot and connect the points in order.

Step 2: Decide what kind of polygon it is.

\(MN\) and \(OP\) are horizontal. \(NO\) and \(PM\) are vertical. Opposite sides are equal, and all angles are right angles, so the polygon is a rectangle.

Step 3: Find side lengths.

$$MN = |6-1| = 5$$

$$NO = |7-3| = 4$$

So the perimeter is:

$$5+4+5+4=18$$

Answer: The polygon is a rectangle with perimeter 18 units.

Tips for Success

• Always read coordinates in the correct order: \((x,y)\).
• Move left or right first, then up or down.
• Check whether a side is horizontal or vertical before finding its length.
• Use absolute value so lengths are positive.
• When finding a missing vertex, match x-values for vertical sides and y-values for horizontal sides.
• Make sure the polygon is closed.

Common Mistakes to Avoid

• Mixing up x and y: \((2,5)\) is not the same point as \((5,2)\).
• Counting spaces incorrectly: each grid square usually represents 1 unit.
• Forgetting the last side: perimeter must include every side.
• Subtracting in the wrong direction without absolute value: distance cannot be negative.

Quick Check

1. If points \((2,4)\) and \((7,4)\) are connected, is the side horizontal or vertical?
2. What is the length of the side from \((3,1)\) to \((3,8)\)?
3. What is the missing vertex of a rectangle with vertices \((0,0)\), \((0,5)\), and \((6,5)\)?

Answers:

1. Horizontal, because the y-coordinates are the same.
2. \(|8-1|=7\), so the length is 7 units.
3. The missing vertex is \((6,0)\).

Summary

Polygons on a coordinate plane are made by plotting vertices and connecting them in order. You can use the coordinates to identify horizontal and vertical sides, find side lengths, and calculate perimeter.

When a vertex is missing, look for matching x-coordinates and y-coordinates to complete the shape. With careful plotting and checking, you can draw and analyze polygons correctly.