Finding the Distance Between Two Points

To find the distance between two points in a coordinate plane, we use the distance formula. This formula is derived from the Pythagorean theorem and is essential in coordinate geometry.

Understanding the Distance Formula

The distance formula states that the distance \(d\) between two points with coordinates \((x_1, y_1)\) and \((x_2, y_2)\) is given by:

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

Here, \(x_1\) and \(y_1\) are the coordinates of the first point, and \(x_2\) and \(y_2\) are the coordinates of the second point.

Applying the Formula to the Given Points

We are given two points: \((4, 3)\) and \((3, -2)\).

Let the first point be \((x_1, y_1) = (4, 3)\).

Let the second point be \((x_2, y_2) = (3, -2)\).

Now, we substitute these values into the distance formula:

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

Next, we perform the subtractions inside the parentheses:

• \(3 - 4 = -1\)
• \(-2 - 3 = -5\)

Substitute these results back into the formula:

\[d = \sqrt{(-1)^2 + (-5)^2}\]

Now, square the results:

• \((-1)^2 = 1\)
• \((-5)^2 = 25\)

Substitute the squared values back into the formula:

\[d = \sqrt{1 + 25}\]

Finally, add the numbers under the square root:

\[d = \sqrt{26}\]

Conclusion

The distance between the points \((4, 3)\) and \((3, -2)\) is \(\sqrt{26}\).

Checking the Options

Let's compare our calculated distance with the given options:

• Option 1: \(\sqrt{26}\)
• Option 2: \(\sqrt{24}\)
• Option 3: 6
• Option 4: 5

Our calculated distance, \(\sqrt{26}\), matches Option 1.

Revision Table: Distance Formula Key Points

Additional Information: Visualizing Distance

You can visualize the distance between points \((4, 3)\) and \((3, -2)\) on a coordinate plane. Imagine moving from \((4, 3)\) to \((3, -2)\). The horizontal change is \(3 - 4 = -1\) (1 unit to the left), and the vertical change is \(-2 - 3 = -5\) (5 units down). These changes form the legs of a right triangle with lengths 1 and 5. The distance between the points is the hypotenuse of this triangle. Using the Pythagorean theorem (\(1^2 + 5^2 = d^2\)), we get \(1 + 25 = d^2\), so \(d^2 = 26\), and \(d = \sqrt{26}\). This confirms the distance formula result.

The distance is always a non-negative value, representing a length.