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

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

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

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

### Step 1: Identify the coordinates

\[
(x_1, y_1) = (3, -4), \quad (x_2, y_2) = (-10, -2)
\]

### Step 2: Apply the distance formula

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

### Step 3: Simplify the expressions inside the square root

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

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

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

\[
d = \sqrt{173}
\]

### Conclusion

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

\[
\boxed{\sqrt{173}}
\]

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

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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 \((7, 24)\)?
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.

Learn how to calculate the distance between two points (3, -4) and (-10, -2) on a coordinate plane using the distance formula. This problem involves applying the distance formula step-by-step to find the precise distance between the given points. Suitable for students in Grades 7-9.

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