To find the distance between two points in a coordinate plane, you can use the Distance Formula. This formula is derived from the Pythagorean Theorem and works for points in both two-dimensional (2D) and three-dimensional (3D) space. Here's how you can approach it:

1. Distance Between Two Points in 2D

Consider two points, $A(x_1, y_1)$ and $B(x_2, y_2)$, in a 2D plane.

Step 1: Identify the coordinates of both points.

• Point A: $(x_1, y_1)$
• Point B: $(x_2, y_2)$

**Step 2: **Subtract the coordinates to find the differences:

• Difference in x-coordinates: $x_2 - x_1$
• Difference in y-coordinates: $y_2 - y_1$

Step 3: Apply the Distance Formula:

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

Here, $d$ represents the distance between the two points.

Example:

Find the distance between $A(2, 3)$ and $B(5, 7)$.

$d = \sqrt{(5 - 2)^2 + (7 - 3)^2} = \sqrt{3^2 + 4^2} = \sqrt{9 + 16} = \sqrt{25} = 5$

So, the distance between the points is 5 units.

2. Distance Between Two Points in 3D

If you have points in 3D space, $A(x_1, y_1, z_1)$ and $B(x_2, y_2, z_2)$, you use a similar formula with an additional term for the z-coordinates.

Step 1: Identify the coordinates of both points.

• Point A: $(x_1, y_1, z_1)$
• Point B: $(x_2, y_2, z_2)$

Step 2: Apply the 3D Distance Formula:

$d = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2 + (z_2 - z_1)^2}$

Example:

Find the distance between $A(1, 2, 3)$ and $B(4, 6, 8)$.

$d = \sqrt{(4 - 1)^2 + (6 - 2)^2 + (8 - 3)^2} = \sqrt{3^2 + 4^2 + 5^2} = \sqrt{9 + 16 + 25} = \sqrt{50} \approx 7.07$

So, the distance between the points is approximately 7.07 units.

---

Do you want more details or have any questions?

5 Related Questions:

1. How do you derive the distance formula from the Pythagorean Theorem?
2. How do you find the midpoint between two points?
3. What is the distance between points on a number line?
4. How does the distance formula change if the plane is not Cartesian?
5. Can the distance formula be applied to points in higher dimensions?

Tip:

Always double-check your calculations by squaring the differences carefully to avoid errors when finding the distance.