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Distance Formula in the Coordinate Plane

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Practice Distance Formula in the Coordinate Plane
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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.

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.

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

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)

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.

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

Practice

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)