Math Problem Statement

The distance between (3,−4) and (−10,−2) is

Solution

To find the distance between two points (x1,y1)(x_1, y_1) and (x2,y2)(x_2, y_2) in a coordinate plane, we use the distance formula:

d=(x2x1)2+(y2y1)2d = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2}

Given the points (3,4)(3, -4) and (10,2)(-10, -2), let's substitute these coordinates into the formula.

Step 1: Identify the coordinates

(x1,y1)=(3,4),(x2,y2)=(10,2)(x_1, y_1) = (3, -4), \quad (x_2, y_2) = (-10, -2)

Step 2: Apply the distance formula

d=((10)3)2+((2)(4))2d = \sqrt{((-10) - 3)^2 + ((-2) - (-4))^2}

Step 3: Simplify the expressions inside the square root

d=(103)2+(2+4)2d = \sqrt{(-10 - 3)^2 + (-2 + 4)^2}

d=(13)2+22d = \sqrt{(-13)^2 + 2^2}

d=169+4d = \sqrt{169 + 4}

d=173d = \sqrt{173}

Conclusion

The distance between the points (3,4)(3, -4) and (10,2)(-10, -2) is:

173\boxed{\sqrt{173}}

Would you like more details, or do you have any questions about this solution?


Here are 5 related questions to consider:

  1. How do you derive the distance formula for two points on a coordinate plane?
  2. What is the distance between the origin and the point (7,24)(7, 24)?
  3. How do you find the midpoint between two points?
  4. If a point moves horizontally by 3 units and vertically by 4 units, what is the distance traveled?
  5. How can you find the distance between a point and a line in the coordinate plane?

Tip: When calculating distances in coordinate geometry, always double-check the signs of your coordinate differences to avoid errors.

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Math Problem Analysis

Mathematical Concepts

Coordinate Geometry
Distance Formula

Formulas

Distance formula: d = sqrt((x2 - x1)^2 + (y2 - y1)^2)

Theorems

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Suitable Grade Level

Grades 7-9