distance on the coordinate plane unit pythagorean theorem homework 4

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Pythagorean Theorem On Coordinate Systems Worksheets

How to Use the Pythagorean Theorem on Coordinate Systems - Finding the length using Pythagoras theorem on a coordinate system can sometimes sound difficult, but it is very easy. The majority of the students are familiar with the concept of Pythagoras theorem, where the square on the hypotenuse is equal to the sum of the square of the remaining two sides of the same triangle. The coordinate system can be used to make sure that the lengths of your known sides of the triangle can be easily found. If a and b are the legs of the triangle that c is the hypotenuse and the theorem will be presented as: a 2 + b 2 = c 2

Aligned Standard: Grade 8 Geometry - 8.G.B.8

• Finding Distance Step-by-Step Lesson - Find the distance between two points by using triangle theory.
• Guided Lesson - Find the distance of points that are plotted over all kinds of quadrants.
• Guided Lesson Explanation - By the end of this unit you will be Pythagorean experts; at least when working with coordinate systems.
• Independent Practice - 8 practice problems that will take about 5 minutes each. They are spread over 4 pages.
• Matching Worksheet - Find the distances that match the graphs that we present you with.
• Distance Formula Worksheet Five Pack - You are basically looking to see how far two things are apart.
• Answer Keys - These are for all the unlocked materials above.

Homework Sheets

The coordinate graph makes it much more understandable than just labeling sides.

• Homework 1 - Apply the Pythagorean Theorem to find the distance between two points in a coordinate system.
• Homework 2 - Create your own little triangle to get this one done.
• Homework 3 - Find the distance between (1, 4) and (3, -4).

Practice Worksheets

We start to find random distances between points.

• Practice 1 - More points and the distances that keep them apart.
• Practice 2 - The lines rise and fall.
• Practice 3 - Use Pythagorean Theorem we count the column and given the value of a and b from find the length of two points a and b.

Math Skill Quizzes

Here are more like problems to help you master this topic.

• Quiz 1 - Straight up problems, literally.
• Quiz 2 - This is how submarine engineers locate objects in the water.
• Quiz 3 - Find the distance between (3, 0) and (-3,2).

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Module 3: The Rectangular Coordinate System and Equations of Lines

Distance in the coordinate plane, learning outcomes.

• Use the distance formula to find the distance between two points in the plane.
• Use the midpoint formula to find the midpoint between two points.

Derived from the Pythagorean Theorem , the distance formula is used to find the distance between two points in the plane. The Pythagorean Theorem, [latex]{a}^{2}+{b}^{2}={c}^{2}[/latex], is based on a right triangle where a and b are the lengths of the legs adjacent to the right angle, and c is the length of the hypotenuse.

The relationship of sides [latex]|{x}_{2}-{x}_{1}|[/latex] and [latex]|{y}_{2}-{y}_{1}|[/latex] to side d is the same as that of sides a and b to side c. We use the absolute value symbol to indicate that the length is a positive number because the absolute value of any number is positive. (For example, [latex]|-3|=3[/latex]. ) The symbols [latex]|{x}_{2}-{x}_{1}|[/latex] and [latex]|{y}_{2}-{y}_{1}|[/latex] indicate that the lengths of the sides of the triangle are positive. To find the length c , take the square root of both sides of the Pythagorean Theorem.

It follows that the distance formula is given as

We do not have to use the absolute value symbols in this definition because any number squared is positive.

A General Note: The Distance Formula

Given endpoints [latex]\left({x}_{1},{y}_{1}\right)[/latex] and [latex]\left({x}_{2},{y}_{2}\right)[/latex], the distance between two points is given by

Example: Finding the Distance between Two Points

Find the distance between the points [latex]\left(-3,-1\right)[/latex] and [latex]\left(2,3\right)[/latex].

Let us first look at the graph of the two points. Connect the points to form a right triangle.

Then, calculate the length of d using the distance formula.

Find the distance between two points: [latex]\left(1,4\right)[/latex] and [latex]\left(11,9\right)[/latex].

[latex]\sqrt{125}=5\sqrt{5}[/latex]

In the following video, we present more worked examples of how to use the distance formula to find the distance between two points in the coordinate plane.

Example: Finding the Distance between Two Locations

Let’s return to the situation introduced at the beginning of this section.

Tracie set out from Elmhurst, IL to go to Franklin Park. On the way, she made a few stops to do errands. Each stop is indicated by a red dot. Find the total distance that Tracie traveled. Compare this with the distance between her starting and final positions.

The first thing we should do is identify ordered pairs to describe each position. If we set the starting position at the origin, we can identify each of the other points by counting units east (right) and north (up) on the grid. For example, the first stop is 1 block east and 1 block north, so it is at [latex]\left(1,1\right)[/latex]. The next stop is 5 blocks to the east so it is at [latex]\left(5,1\right)[/latex]. After that, she traveled 3 blocks east and 2 blocks north to [latex]\left(8,3\right)[/latex]. Lastly, she traveled 4 blocks north to [latex]\left(8,7\right)[/latex]. We can label these points on the grid.

Next, we can calculate the distance. Note that each grid unit represents 1,000 feet.

• From her starting location to her first stop at [latex]\left(1,1\right)[/latex], Tracie might have driven north 1,000 feet and then east 1,000 feet, or vice versa. Either way, she drove 2,000 feet to her first stop.
• Her second stop is at [latex]\left(5,1\right)[/latex]. So from [latex]\left(1,1\right)[/latex] to [latex]\left(5,1\right)[/latex], Tracie drove east 4,000 feet.
• Her third stop is at [latex]\left(8,3\right)[/latex]. There are a number of routes from [latex]\left(5,1\right)[/latex] to [latex]\left(8,3\right)[/latex]. Whatever route Tracie decided to use, the distance is the same, as there are no angular streets between the two points. Let’s say she drove east 3,000 feet and then north 2,000 feet for a total of 5,000 feet.
• Tracie’s final stop is at [latex]\left(8,7\right)[/latex]. This is a straight drive north from [latex]\left(8,3\right)[/latex] for a total of 4,000 feet.

Next, we will add the distances listed in the table.

The total distance Tracie drove is 15,000 feet or 2.84 miles. This is not, however, the actual distance between her starting and ending positions. To find this distance, we can use the distance formula between the points [latex]\left(0,0\right)[/latex] and [latex]\left(8,7\right)[/latex].

At 1,000 feet per grid unit, the distance between Elmhurst, IL to Franklin Park is 10,630.14 feet, or 2.01 miles. The distance formula results in a shorter calculation because it is based on the hypotenuse of a right triangle, a straight diagonal from the origin to the point [latex]\left(8,7\right)[/latex]. Perhaps you have heard the saying “as the crow flies,” which means the shortest distance between two points because a crow can fly in a straight line even though a person on the ground has to travel a longer distance on existing roadways.

Using the Midpoint Formula

When the endpoints of a line segment are known, we can find the point midway between them. This point is known as the midpoint and the formula is known as the midpoint formula . Given the endpoints of a line segment, [latex]\left({x}_{1},{y}_{1}\right)[/latex] and [latex]\left({x}_{2},{y}_{2}\right)[/latex], the midpoint formula states how to find the coordinates of the midpoint [latex]M[/latex].

A graphical view of a midpoint is shown below. Notice that the line segments on either side of the midpoint are congruent.

Example: Finding the Midpoint of the Line Segment

Find the midpoint of the line segment with the endpoints [latex]\left(7,-2\right)[/latex] and [latex]\left(9,5\right)[/latex].

Use the formula to find the midpoint of the line segment.

Find the midpoint of the line segment with endpoints [latex]\left(-2,-1\right)[/latex] and [latex]\left(-8,6\right)[/latex].

[latex]\left(-5,\frac{5}{2}\right)[/latex]

Example: Finding the Center of a Circle

The diameter of a circle has endpoints [latex]\left(-1,-4\right)[/latex] and [latex]\left(5,-4\right)[/latex]. Find the center of the circle.

The center of a circle is the center or midpoint of its diameter. Thus, the midpoint formula will yield the center point.

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Course: class 10 (old)   >   unit 7, finding distance with pythagorean theorem.

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Video transcript

Coordinate Distance Calculator

What is the distance formula for cartesian coordinates, how to use the coordinate distance calculator.

• Let's keep finding distances!

Use the coordinate distance calculator to find the distance between two coordinates in a two-dimensional or three-dimensional space. By simply entering the XY or XYZ coordinates of the points, this tool will instantly compute the distance between them!

Along with this tool, we've created a brief text where you'll find:

• What the distance formula is for cartesian coordinates ;
• How to use this formula for determining the distance between coordinates in the 2D or 3D spaces ; and
• The distance formula for polar coordinates .

The general distance formula in cartesian coordinates is:

d = √[(x₂ - x₁)² + (y₂ - y₁)² + (z₂ - z₁)²]

• d — Distance between two coordinates;
• x₁ , y₁ and z₁ — 3D coordinates of any of the points; and
• x₂ , y₂ and z₂ — 3D coordinates of the other point.

This formula, which derives from the Pythagorean theorem, is also known as the Euclidian distance formula for three-dimensional space.

Although this formula includes the z coordinate, you may use it for both 2D and 3D spaces. By setting the z coordinates to zero, you can get a particular version for the distance between two points in a 2D space:

d = √[(x₂ - x₁)² + (y₂ - y₁)²]

Here d is the distance between two points in the two-dimensional space.

The coordinate distance calculator makes it simple to find the distance between two points given its cartesian coordinates. Let us see how to use this tool:

From the Dimensions field, choose between 2D or 3D , according to the dimensional space in which your points are defined.

In the First point section of the calculator, enter the coordinates of one of the points.

Similarly, in the Second point section, input the coordinates' values of the other point.

Once you've entered these values, the calculator will display the distance between the points ( Distance ) in the Result section.

🙋 Did you know that a 2D coordinate can be expressed as a 3D point that has a z coordinate equal to zero (x, y, 0)? This means you could as well use the 3D version of this coordinate distance calculator to find distances between 2D points by simply setting the z coordinates to zero.

Let's keep finding distances!

Now that you've mastered how to calculate the distance between two coordinates, you might want to take a look at some other related tools:

• Distance calculator ;
• 2D distance calculator ;
• Length of a line segment calculator ;
• Euclidean distance calculator ; and
• Distance between two points calculator .

What is the 3D distance formula?

The 3D distance formula is:

• d — Distance between the two coordinates;
• x₁ , y₁ and z₁ — 3D coordinates of point one; and
• x₂ , y₂ and z₂ — 3D coordinates of point two.

How do I calculate the distance between the two coordinates?

To find the distance between two three-dimensional coordinates (-1, 0, 2) and (3, 5, 4):

Use the distance formula for 3D coordinates:

d = √[(x₂ - x₁)² + (y₂ - y₁)²+ (z₂ - z₁)²]

The variable's values from that equation are:

(x₁, y₁, z₁) = (-1, 0, 2) (x₂, y₂, z₂) = (3, 5, 4)

Substitute and perform the corresponding calculations:

d = √[(3 - -1)² + (5 - 0)² + (4 - 2)²] d = √[(4)² + (5)² + (2)²] d = √45

Calculate the square root to get the distance: d = 6.70825

Can I calculate the distance between polar coordinates?

Yes. By employing the distance formula for polar coordinates d = √[r₁² + r₂² - 2r₁r₂cos(θ₁ - θ₂)] , you can determine the distance between two polar coordinates in a two-dimensional or three-dimensional space.

What is the distance formula in polar coordinates?

The distance formula for polar coordinates is:

d = √[r₁² + r₂² - 2r₁r₂cos(θ₁ - θ₂)]

• d — Distance between the two points;
• r₁ and θ₁ — Polar coordinates of point one; and
• r₂ and θ₂ — Polar coordinates of point two.

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38.4: Distances on a Coordinate Plane

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Let's explore distance on the coordinate plane.

Exercise \(\PageIndex{1}\): Coordinate Patterns

Plot points in your assigned quadrant and label them with their coordinates.

Exercise \(\PageIndex{2}\): Signs of Numbers in Coordinates

• Write the coordinates of each point.
• Answer these questions for each pair of points.
• How are the coordinates the same? How are they different?
• How far away are they from the y-axis? To the left or to the right of it?
• How far away are they from the x-axis? Above or below it?
• \(A\) and \(B\)
• \(B\) and \(D\)
• \(A\) and \(D\)

Pause here for a class discussion.

• Point \(F\) has the same coordinates as point \(C\), except its \(y\)-coordinate has the opposite sign.
• Plot point \(F\) on the coordinate plane and label it with its coordinates.
• How far away are \(F\) and \(C\) from the \(x\)-axis?
• What is the distance between \(F\) and \(C\)?
• Plot point \(G\) on the coordinate plane and label it with its coordinates.
• How far away are \(G\) and \(E\) from the \(y\)-axis?
• What is the distance between and ?
• Point \(H\) has the same coordinates as point \(B\), except both of its coordinates have the opposite signs. In which quadrant is point \(H\)?

Exercise \(\PageIndex{3}\): Finding Distances on a Coordinate Plane

• Label each point with its coordinates.
• Point \(B\) and \(C\)
• Point \(D\) and \(B\)
• Point \(D\) and \(E\)
• Which of the points are 5 units from \((-1.5,-3)\)?
• Which of the points are 2 units from \((0.5,-4.5)\)?
• Plot a point that is both 2.5 units from \(A\) and 9 units from \(E\). Label that point \(F\) and write down its coordinates.

Are you ready for more?

Priya says, “There are exactly four points that are 3 units away from \((-5,0)\).” Lin says, “I think there are a whole bunch of points that are 3 units away from \((-5,0)\).”

Do you agree with either of them? Explain your reasoning.

The points \(A=(5,2), B=(-5,2), C=(-5,-2),\) and \(D=(5,-2)\) are shown in the plane. Notice that they all have almost the same coordinates, except the signs are different. They are all the same distance from each axis but are in different quadrants.

Notice that the vertical distance between points \(A\) and \(D\) is 4 units, because point \(A\) is 2 units above the horizontal axis and point \(D\) is 2 units below the horizontal axis. The horizontal distance between points \(A\) and \(B\) is 10 units, because point \(B\) is 5 units to the left of the vertical axis and point \(A\) is 5 units to the right of the vertical axis.

We can always tell which quadrant a point is located in by the signs of its coordinates.

In general:

• If two points have \(x\)-coordinates that are opposites (like 5 and -5), they are the same distance away from the vertical axis, but one is to the left and the other to the right.
• If two points have \(y\)-coordinates that are opposites (like 2 and -2), they are the same distance away from the horizontal axis, but one is above and the other below.

When two points have the same value for the first or second coordinate, we can find the distance between them by subtracting the coordinates that are different. For example, consider \((1,3)\) and \((5,3)\):

They have the same \(y\)-coordinate. If we subtract the \(x\)-coordinates, we get \(5-1=4\). These points are 4 units apart.

Glossary Entries

Definition: Quadrant

The coordinate plane is divided into 4 regions called quadrants. The quadrants are numbered using Roman numerals, starting in the top right corner.

Exercise \(\PageIndex{4}\)

Here are 4 points on a coordinate plane.

• Plot a point that is 3 units from point \(K\). Label it \(P\).
• Plot a point that is 2 units from point \(M\). Label it \(W\).

Exercise \(\PageIndex{5}\)

Each set of points are connected to form a line segment. What is the length of each?

• \(A=(3,5)\) and \(B=(3,6)\)
• \(C=(-2,-3)\) and \(D=(-2,-6)\)
• \(E=(-3,1)\) and \(F=(-3,-1)\)

Exercise \(\PageIndex{6}\)

On the coordinate plane, plot four points that are each 3 units away from point \(P=(-2,-1)\). Write the coordinates of each point.

Exercise \(\PageIndex{7}\)

Noah’s recipe for sparkling orange juice uses 4 liters of orange juice and 5 liters of soda water.

• Noah prepares large batches of sparkling orange juice for school parties. He usually knows the total number of liters, \(t\), that he needs to prepare. Write an equation that shows how Noah can find \(s\), the number of liters of soda water, if he knows \(t\).
• Sometimes the school purchases a certain number, \(j\), of liters of orange juice and Noah needs to figure out how much sparkling orange juice he can make. Write an equation that Noah can use to find \(t\) if he knows \(j\).

(From Unit 6.4.1)

Exercise \(\PageIndex{8}\)

For a suitcase to be checked on a flight (instead of carried by hand), it can weigh at most 50 pounds. Andre’s suitcase weighs 23 kilograms. Can Andre check his suitcase? Explain or show your reasoning. (Note: 10 kilograms \(\approx\) 22 pounds)

(From Unit 3.2.3)

Unit: Pythagorean Theorem Name_ Homework 4 Date _Pd_ DISTANCE ON THE COORDINATE PLANE For questions 1-4, find the distance between points A and B. Round your solutions to the nearest fenth when necessary. _ _ 3 Point A(3,6); Point B(16,16) Point A(7,17); Point B(19,2) _ _ (5) Find the perimeter of the trapezoid. _ Mansuvering the Middle LC, 201

Unit 9 Lesson 4 Homework (Distance on Coordinate Plane)