Apply the Pythagorean Theorem to Find the Distance Between Two Points in a Coordinate System: CCSS.Math.Content.8.G.B.8 - 8th Grade Math
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An airplane is 8 miles west and 15 miles south of its destination. Approximately how far is the plane from its destination, in miles?
An airplane is 8 miles west and 15 miles south of its destination. Approximately how far is the plane from its destination, in miles?
A right triangle can be drawn between the airplane and its destination.
Destination
15 miles 
Airplane
8 miles
We can solve for the hypotenuse, x, of the triangle:
82 + 152 = x2
64 + 225 = x2
289 = x2
x = 17 miles
A right triangle can be drawn between the airplane and its destination.
Destination
15 miles Airplane
8 miles
We can solve for the hypotenuse, x, of the triangle:
82 + 152 = x2
64 + 225 = x2
289 = x2
x = 17 miles
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To get from his house to the hardware store, Bob must drive 3 miles to the east and then 4 miles to the north. If Bob was able to drive along a straight line directly connecting his house to the store, how far would he have to travel then?
To get from his house to the hardware store, Bob must drive 3 miles to the east and then 4 miles to the north. If Bob was able to drive along a straight line directly connecting his house to the store, how far would he have to travel then?
Since east and north directions are perpendicular, the possible routes Bob can take can be represented by a right triangle with sides a and b of length 3 miles and 5 miles, respectively. The hypotenuse c represents the straight line connecting his house to the store, and its length can be found using the Pythagorean theorem: _c_2 = 32+ 42 = 25. Since the square root of 25 is 5, the length of the hypotenuse is 5 miles.
Since east and north directions are perpendicular, the possible routes Bob can take can be represented by a right triangle with sides a and b of length 3 miles and 5 miles, respectively. The hypotenuse c represents the straight line connecting his house to the store, and its length can be found using the Pythagorean theorem: _c_2 = 32+ 42 = 25. Since the square root of 25 is 5, the length of the hypotenuse is 5 miles.
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Use the Pythagorean Theorem to calculate the length of the line shown on the provided coordinate plane. Round the answer to the nearest tenth.
Use the Pythagorean Theorem to calculate the length of the line shown on the provided coordinate plane. Round the answer to the nearest tenth.
Notice that the diagonal line from the problem could be the hypotenuse of a right triangle. If we add two more lines, then we can create a closed figure in the shape of a triangle:
Let's use the Pythagorean Theorem to calculate the length of the line that represents the hypotenuse of a right triangle. The Pythagorean Theorem states that for right triangles:
In this equation:
We can count the number of units on the coordinate plane that were used to create the legs of our drawn triangle. Afterwards, we can use the Pythagorean Theorem to solve for the length of the hypotenuse, or the original diagonal line.
In order to solve for this problem we want to substitute in the known side lengths for the triangle's legs:
Notice that the diagonal line from the problem could be the hypotenuse of a right triangle. If we add two more lines, then we can create a closed figure in the shape of a triangle:
Let's use the Pythagorean Theorem to calculate the length of the line that represents the hypotenuse of a right triangle. The Pythagorean Theorem states that for right triangles:
In this equation:
We can count the number of units on the coordinate plane that were used to create the legs of our drawn triangle. Afterwards, we can use the Pythagorean Theorem to solve for the length of the hypotenuse, or the original diagonal line.
In order to solve for this problem we want to substitute in the known side lengths for the triangle's legs:
Compare your answer with the correct one above
Use the Pythagorean Theorem to calculate the length of the line shown on the provided coordinate plane. Round the answer to the nearest tenth.
Use the Pythagorean Theorem to calculate the length of the line shown on the provided coordinate plane. Round the answer to the nearest tenth.
Notice that the diagonal line from the problem could be the hypotenuse of a right triangle. If we add two more lines, then we can create a closed figure in the shape of a triangle:
Let's use the Pythagorean Theorem to calculate the length of the line that represents the hypotenuse of a right triangle. The Pythagorean Theorem states that for right triangles:
In this equation:
We can count the number of units on the coordinate plane that were used to create the legs of our drawn triangle. Afterwards, we can use the Pythagorean Theorem to solve for the length of the hypotenuse, or the original diagonal line.
In order to solve for this problem we want to substitute in the known side lengths for the triangle's legs:
Notice that the diagonal line from the problem could be the hypotenuse of a right triangle. If we add two more lines, then we can create a closed figure in the shape of a triangle:
Let's use the Pythagorean Theorem to calculate the length of the line that represents the hypotenuse of a right triangle. The Pythagorean Theorem states that for right triangles:
In this equation:
We can count the number of units on the coordinate plane that were used to create the legs of our drawn triangle. Afterwards, we can use the Pythagorean Theorem to solve for the length of the hypotenuse, or the original diagonal line.
In order to solve for this problem we want to substitute in the known side lengths for the triangle's legs:
Compare your answer with the correct one above
Use the Pythagorean Theorem to calculate the length of the line shown on the provided coordinate plane. Round the answer to the nearest tenth.
Use the Pythagorean Theorem to calculate the length of the line shown on the provided coordinate plane. Round the answer to the nearest tenth.
Notice that the diagonal line from the problem could be the hypotenuse of a right triangle. If we add two more lines, then we can create a closed figure in the shape of a triangle:
Let's use the Pythagorean Theorem to calculate the length of the line that represents the hypotenuse of a right triangle. The Pythagorean Theorem states that for right triangles:
In this equation:
We can count the number of units on the coordinate plane that were used to create the legs of our drawn triangle. Afterwards, we can use the Pythagorean Theorem to solve for the length of the hypotenuse, or the original diagonal line.
In order to solve for this problem we want to substitute in the known side lengths for the triangle's legs:
Notice that the diagonal line from the problem could be the hypotenuse of a right triangle. If we add two more lines, then we can create a closed figure in the shape of a triangle:
Let's use the Pythagorean Theorem to calculate the length of the line that represents the hypotenuse of a right triangle. The Pythagorean Theorem states that for right triangles:
In this equation:
We can count the number of units on the coordinate plane that were used to create the legs of our drawn triangle. Afterwards, we can use the Pythagorean Theorem to solve for the length of the hypotenuse, or the original diagonal line.
In order to solve for this problem we want to substitute in the known side lengths for the triangle's legs:
Compare your answer with the correct one above
Use the Pythagorean Theorem to calculate the length of the line shown on the provided coordinate plane. Round the answer to the nearest tenth.
Use the Pythagorean Theorem to calculate the length of the line shown on the provided coordinate plane. Round the answer to the nearest tenth.
Notice that the diagonal line from the problem could be the hypotenuse of a right triangle. If we add two more lines, then we can create a closed figure in the shape of a triangle:
Let's use the Pythagorean Theorem to calculate the length of the line that represents the hypotenuse of a right triangle. The Pythagorean Theorem states that for right triangles:
In this equation:
We can count the number of units on the coordinate plane that were used to create the legs of our drawn triangle. Afterwards, we can use the Pythagorean Theorem to solve for the length of the hypotenuse, or the original diagonal line.
In order to solve for this problem we want to substitute in the known side lengths for the triangle's legs:
Notice that the diagonal line from the problem could be the hypotenuse of a right triangle. If we add two more lines, then we can create a closed figure in the shape of a triangle:
Let's use the Pythagorean Theorem to calculate the length of the line that represents the hypotenuse of a right triangle. The Pythagorean Theorem states that for right triangles:
In this equation:
We can count the number of units on the coordinate plane that were used to create the legs of our drawn triangle. Afterwards, we can use the Pythagorean Theorem to solve for the length of the hypotenuse, or the original diagonal line.
In order to solve for this problem we want to substitute in the known side lengths for the triangle's legs:
Compare your answer with the correct one above
Use the Pythagorean Theorem to calculate the length of the line shown on the provided coordinate plane. Round the answer to the nearest tenth.
Use the Pythagorean Theorem to calculate the length of the line shown on the provided coordinate plane. Round the answer to the nearest tenth.
Notice that the diagonal line from the problem could be the hypotenuse of a right triangle. If we add two more lines, then we can create a closed figure in the shape of a triangle:
Let's use the Pythagorean Theorem to calculate the length of the line that represents the hypotenuse of a right triangle. The Pythagorean Theorem states that for right triangles:
In this equation:
We can count the number of units on the coordinate plane that were used to create the legs of our drawn triangle. Afterwards, we can use the Pythagorean Theorem to solve for the length of the hypotenuse, or the original diagonal line.
In order to solve for this problem we want to substitute in the known side lengths for the triangle's legs:
Notice that the diagonal line from the problem could be the hypotenuse of a right triangle. If we add two more lines, then we can create a closed figure in the shape of a triangle:
Let's use the Pythagorean Theorem to calculate the length of the line that represents the hypotenuse of a right triangle. The Pythagorean Theorem states that for right triangles:
In this equation:
We can count the number of units on the coordinate plane that were used to create the legs of our drawn triangle. Afterwards, we can use the Pythagorean Theorem to solve for the length of the hypotenuse, or the original diagonal line.
In order to solve for this problem we want to substitute in the known side lengths for the triangle's legs:
Compare your answer with the correct one above
Use the Pythagorean Theorem to calculate the length of the line shown on the provided coordinate plane. Round the answer to the nearest tenth.
Use the Pythagorean Theorem to calculate the length of the line shown on the provided coordinate plane. Round the answer to the nearest tenth.
Notice that the diagonal line from the problem could be the hypotenuse of a right triangle. If we add two more lines, then we can create a closed figure in the shape of a triangle:
Let's use the Pythagorean Theorem to calculate the length of the line that represents the hypotenuse of a right triangle. The Pythagorean Theorem states that for right triangles:
In this equation:
We can count the number of units on the coordinate plane that were used to create the legs of our drawn triangle. Afterwards, we can use the Pythagorean Theorem to solve for the length of the hypotenuse, or the original diagonal line.
In order to solve for this problem we want to substitute in the known side lengths for the triangle's legs:
Notice that the diagonal line from the problem could be the hypotenuse of a right triangle. If we add two more lines, then we can create a closed figure in the shape of a triangle:
Let's use the Pythagorean Theorem to calculate the length of the line that represents the hypotenuse of a right triangle. The Pythagorean Theorem states that for right triangles:
In this equation:
We can count the number of units on the coordinate plane that were used to create the legs of our drawn triangle. Afterwards, we can use the Pythagorean Theorem to solve for the length of the hypotenuse, or the original diagonal line.
In order to solve for this problem we want to substitute in the known side lengths for the triangle's legs:
Compare your answer with the correct one above
Use the Pythagorean Theorem to calculate the length of the line shown on the provided coordinate plane. Round the answer to the nearest tenth.
Use the Pythagorean Theorem to calculate the length of the line shown on the provided coordinate plane. Round the answer to the nearest tenth.
Notice that the diagonal line from the problem could be the hypotenuse of a right triangle. If we add two more lines, then we can create a closed figure in the shape of a triangle:
Let's use the Pythagorean Theorem to calculate the length of the line that represents the hypotenuse of a right triangle. The Pythagorean Theorem states that for right triangles:
In this equation:
We can count the number of units on the coordinate plane that were used to create the legs of our drawn triangle. Afterwards, we can use the Pythagorean Theorem to solve for the length of the hypotenuse, or the original diagonal line.
In order to solve for this problem we want to substitute in the known side lengths for the triangle's legs:
Notice that the diagonal line from the problem could be the hypotenuse of a right triangle. If we add two more lines, then we can create a closed figure in the shape of a triangle:
Let's use the Pythagorean Theorem to calculate the length of the line that represents the hypotenuse of a right triangle. The Pythagorean Theorem states that for right triangles:
In this equation:
We can count the number of units on the coordinate plane that were used to create the legs of our drawn triangle. Afterwards, we can use the Pythagorean Theorem to solve for the length of the hypotenuse, or the original diagonal line.
In order to solve for this problem we want to substitute in the known side lengths for the triangle's legs:
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Angela drives 30 miles north and then 40 miles east. How far is she from where she began?
Angela drives 30 miles north and then 40 miles east. How far is she from where she began?
By drawing Angela’s route, we can connect her end point and her start point with a straight line and will then have a right triangle. The Pythagorean theorem can be used to solve for how far she is from the starting point: a2+b2=c2, 302+402=c2, c=50. It can also be noted that Angela’s route represents a multiple of the 3-4-5 Pythagorean triple.
By drawing Angela’s route, we can connect her end point and her start point with a straight line and will then have a right triangle. The Pythagorean theorem can be used to solve for how far she is from the starting point: a2+b2=c2, 302+402=c2, c=50. It can also be noted that Angela’s route represents a multiple of the 3-4-5 Pythagorean triple.
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In order to get to work, Jeff leaves home and drives 4 miles due north, then 3 miles due east, followed by 6 miles due north and, finally, 7 miles due east. What is the straight line distance from Jeff’s work to his home?
In order to get to work, Jeff leaves home and drives 4 miles due north, then 3 miles due east, followed by 6 miles due north and, finally, 7 miles due east. What is the straight line distance from Jeff’s work to his home?
Jeff drives a total of 10 miles north and 10 miles east. Using the Pythagorean theorem (a2+b2=c2), the direct route from Jeff’s home to his work can be calculated. 102+102=c2. 200=c2. √200=c. √100√2=c. 10√2=c
Jeff drives a total of 10 miles north and 10 miles east. Using the Pythagorean theorem (a2+b2=c2), the direct route from Jeff’s home to his work can be calculated. 102+102=c2. 200=c2. √200=c. √100√2=c. 10√2=c
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Jim leaves his home and walks 10 minutes due west and 5 minutes due south. If Jim could walk a straight line from his current position back to his house, how far, in minutes, is Jim from home?
Jim leaves his home and walks 10 minutes due west and 5 minutes due south. If Jim could walk a straight line from his current position back to his house, how far, in minutes, is Jim from home?
By using Pythagorean Theorem, we can solve for the distance “as the crow flies” from Jim to his home:
102 + 52 = _x_2
100 + 25 = _x_2
√125 = x, but we still need to factor the square root
√125 = √25*5, and since the √25 = 5, we can move that outside of the radical, so
5√5= x
By using Pythagorean Theorem, we can solve for the distance “as the crow flies” from Jim to his home:
102 + 52 = _x_2
100 + 25 = _x_2
√125 = x, but we still need to factor the square root
√125 = √25*5, and since the √25 = 5, we can move that outside of the radical, so
5√5= x
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Justin travels 
to the east and 
to the north. How far away from his starting point is he now?
Justin travels
This is solving for the hypotenuse of a triangle. Using the Pythagorean Theorem, which says that 
This is solving for the hypotenuse of a triangle. Using the Pythagorean Theorem, which says that
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You leave on a road trip driving due North from Savannah, Georgia, at 8am. You drive for 5 hours at 60mph and then head due East for 2 hours at 50mph. After those 7 hours, how far are you Northeast from Savannah as the crow flies (in miles)?
You leave on a road trip driving due North from Savannah, Georgia, at 8am. You drive for 5 hours at 60mph and then head due East for 2 hours at 50mph. After those 7 hours, how far are you Northeast from Savannah as the crow flies (in miles)?
Distance = hours * mph
North Distance = 5 hours * 60 mph = 300 miles
East Distance = 2 hours * 50 mph = 100 miles
Use Pythagorean Theorem to determine Northeast Distance
3002 + 1002 =NE2
90000 + 10000 = 100000 = NE2
NE = √100000
Distance = hours * mph
North Distance = 5 hours * 60 mph = 300 miles
East Distance = 2 hours * 50 mph = 100 miles
Use Pythagorean Theorem to determine Northeast Distance
3002 + 1002 =NE2
90000 + 10000 = 100000 = NE2
NE = √100000
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A park is designed to fit within the confines of a triangular lot in the middle of a city. The side that borders Elm street is 15 feet long. The side that borders Broad street is 23 feet long. Elm street and Broad street meet at a right angle. The third side of the park borders Popeye street, what is the length of the side of the park that borders Popeye street?
A park is designed to fit within the confines of a triangular lot in the middle of a city. The side that borders Elm street is 15 feet long. The side that borders Broad street is 23 feet long. Elm street and Broad street meet at a right angle. The third side of the park borders Popeye street, what is the length of the side of the park that borders Popeye street?
This question requires the use of Pythagorean Theorem. We are given the length of two sides of a triangle and asked to find the third. We are told that the two sides we are given meet at a right angle, this means that the missing side is the hypotenuse. So we use a2 + b2 = c2, plugging in the two known lengths for a and b. This yields an answer of 27.46 feet.
This question requires the use of Pythagorean Theorem. We are given the length of two sides of a triangle and asked to find the third. We are told that the two sides we are given meet at a right angle, this means that the missing side is the hypotenuse. So we use a2 + b2 = c2, plugging in the two known lengths for a and b. This yields an answer of 27.46 feet.
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Daria and Ashley start at the same spot and walk their two dogs to the park, taking different routes. Daria walks 1 mile north and then 1 mile east. Ashley walks her dog on a path going northeast that leads directly to the park. How much further does Daria walk than Ashley?
Daria and Ashley start at the same spot and walk their two dogs to the park, taking different routes. Daria walks 1 mile north and then 1 mile east. Ashley walks her dog on a path going northeast that leads directly to the park. How much further does Daria walk than Ashley?
First let's calculate how far Daria walks. This is simply 1 mile north + 1 mile east = 2 miles. Now let's calculate how far Ashley walks. We can think of this problem using a right triangle. The two legs of the triangle are the 1 mile north and 1 mile east, and Ashley's distance is the diagonal. Using the Pythagorean Theorem we calculate the diagonal as √(12 + 12) = √2. So Daria walked 2 miles, and Ashley walked √2 miles. Therefore the difference is simply 2 – √2 miles.
First let's calculate how far Daria walks. This is simply 1 mile north + 1 mile east = 2 miles. Now let's calculate how far Ashley walks. We can think of this problem using a right triangle. The two legs of the triangle are the 1 mile north and 1 mile east, and Ashley's distance is the diagonal. Using the Pythagorean Theorem we calculate the diagonal as √(12 + 12) = √2. So Daria walked 2 miles, and Ashley walked √2 miles. Therefore the difference is simply 2 – √2 miles.
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Max starts at Point A and travels 6 miles north to Point B and then 4 miles east to Point C. What is the shortest distance from Point A to Point C?
Max starts at Point A and travels 6 miles north to Point B and then 4 miles east to Point C. What is the shortest distance from Point A to Point C?
This can be solved with the Pythagorean Theorem.
62 + 42 = _c_2
52 = _c_2
c = √52 = 2√13
This can be solved with the Pythagorean Theorem.
62 + 42 = _c_2
52 = _c_2
c = √52 = 2√13
Compare your answer with the correct one above
Use the Pythagorean Theorem to calculate the length of the line shown on the provided coordinate plane. Round the answer to the nearest tenth.
Use the Pythagorean Theorem to calculate the length of the line shown on the provided coordinate plane. Round the answer to the nearest tenth.
Notice that the diagonal line from the problem could be the hypotenuse of a right triangle. If we add two more lines, then we can create a closed figure in the shape of a triangle:
Let's use the Pythagorean Theorem to calculate the length of the line that represents the hypotenuse of a right triangle. The Pythagorean Theorem states that for right triangles:
In this equation:
We can count the number of units on the coordinate plane that were used to create the legs of our drawn triangle. Afterwards, we can use the Pythagorean Theorem to solve for the length of the hypotenuse, or the original diagonal line.
In order to solve for this problem we want to substitute in the known side lengths for the triangle's legs:
Notice that the diagonal line from the problem could be the hypotenuse of a right triangle. If we add two more lines, then we can create a closed figure in the shape of a triangle:
Let's use the Pythagorean Theorem to calculate the length of the line that represents the hypotenuse of a right triangle. The Pythagorean Theorem states that for right triangles:
In this equation:
We can count the number of units on the coordinate plane that were used to create the legs of our drawn triangle. Afterwards, we can use the Pythagorean Theorem to solve for the length of the hypotenuse, or the original diagonal line.
In order to solve for this problem we want to substitute in the known side lengths for the triangle's legs:
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Kathy and Jill are travelling from their home to the same destination. Kathy travels due east and then after travelling 6 miles turns and travels 8 miles due north. Jill travels directly from her home to the destination. How miles does Jill travel?
Kathy and Jill are travelling from their home to the same destination. Kathy travels due east and then after travelling 6 miles turns and travels 8 miles due north. Jill travels directly from her home to the destination. How miles does Jill travel?
Kathy's path traces the outline of a right triangle with legs of 6 and 8. By using the Pythagorean Theorem
miles
Kathy's path traces the outline of a right triangle with legs of 6 and 8. By using the Pythagorean Theorem
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Sam and John both start at the same point. Sam walks 30 feet north while John walks 40 feet west. How far apart are they at their new locations?
Sam and John both start at the same point. Sam walks 30 feet north while John walks 40 feet west. How far apart are they at their new locations?
Sam and John have walked at right angles to each other, so the distance between them is the hypotenuse of a triangle. The distance can be found using the Pythagorean Theorem.
Sam and John have walked at right angles to each other, so the distance between them is the hypotenuse of a triangle. The distance can be found using the Pythagorean Theorem.
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