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Distance Formula in the Coordinate Plane

Length between two points using a right triangle.

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Distance Formula in the Coordinate Plane

What if you were given the coordinates of two points? How could you find how far apart these two points are? After completing this Concept, you'll be able to find the distance between two points in the coordinate plane using the Distance Formula.

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CK-12 Finding The Distance Between Two Points

James Sousa: The Distance Formula

Guidance

The distance between two points (x_1, y_1) and (x_2, y_2) can be defined as d= \sqrt{(x_2-x_1)^2 + (y_2-y_1)^2} . This is called the distance formula . Remember that distances are always positive!

Example A

Find the distance between (4, -2) and (-10, 3).

Plug in (4, -2) for (x_1, y_1) and (-10, 3) for (x_2, y_2) and simplify.

d& = \sqrt{(-10-4)^2+(3+2)^2}\\& = \sqrt{(-14)^2 + (5)^2}\\& = \sqrt{196+25}\\& = \sqrt{221} \approx 14.87 \ units

Example B

Find the distance between (3, 4) and (-1, 3).

Plug in (3, 4) for (x_1, y_1) and (-1, 3) for (x_2, y_2) and simplify.

d& = \sqrt{(-1-3)^2+(3-4)^2}\\& = \sqrt{(-4)^2 + (-1)^2}\\& = \sqrt{16+1}\\& = \sqrt{17} \approx 4.12 \ units

Example C

Find the distance between (4, 23) and (8, 14).

Plug in (4, 23) for (x_1, y_1) and (8, 14) for (x_2, y_2) and simplify.

d& = \sqrt{(8-4)^2+(14-23)^2}\\& = \sqrt{(4)^2 + (-9)^2}\\& = \sqrt{16+81}\\& = \sqrt{97} \approx 9.85 \ units

CK-12 Finding The Distance Between Two Points

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Guided Practice

1. Find the distance between (-2, -3) and (3, 9).

2. Find the distance between (12, 26) and (8, 7)

3. Find the distance between (5, 2) and (6, 1)

Answers

1. Use the distance formula, plug in the points, and simplify.

d & = \sqrt{(3-(-2))^2 + (9-(-3))^2}\\& = \sqrt{(5)^2 + (12)^2}\\& = \sqrt{25+144}\\& = \sqrt{169} = 13 \ units

2. Use the distance formula, plug in the points, and simplify.

d & = \sqrt{(8-12)^2 + (7-26)^2}\\& = \sqrt{(-4)^2 + (-19)^2}\\& = \sqrt{16+361}\\& = \sqrt{377} \approx 19.42 \ units

3. Use the distance formula, plug in the points, and simplify.

d & = \sqrt{(6-5)^2 + (1-2)^2}\\& = \sqrt{(1)^2 + (-1)^2}\\& = \sqrt{1+1}\\& = \sqrt{2} = 1.41 \ units

Explore More

Find the distance between each pair of points. Round your answer to the nearest hundredth.

  1. (4, 15) and (-2, -1)
  2. (-6, 1) and (9, -11)
  3. (0, 12) and (-3, 8)
  4. (-8, 19) and (3, 5)
  5. (3, -25) and (-10, -7)
  6. (-1, 2) and (8, -9)
  7. (5, -2) and (1, 3)
  8. (-30, 6) and (-23, 0)
  9. (2, -2) and (2, 5)
  10. (-9, -4) and (1, -1)

Vocabulary

Distance Formula

Distance Formula

The distance between two points (x_1, y_1) and (x_2, y_2) can be defined as d= \sqrt{(x_2-x_1)^2 + (y_2-y_1)^2}.
Pythagorean Theorem

Pythagorean Theorem

The Pythagorean Theorem is a mathematical relationship between the sides of a right triangle, given by a^2 + b^2 = c^2, where a and b are legs of the triangle and c is the hypotenuse of the triangle.

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