(-5/8, -2/5), (-1/4, -5/8) find the distance between the points

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

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

Given the points:
\[
(x_1, y_1) = \left(-\frac{5}{8}, -\frac{2}{5}\right), \quad (x_2, y_2) = \left(-\frac{1}{4}, -\frac{5}{8}\right)
\]

### Step 1: Subtract the \(x\)-coordinates:
\[
x_2 - x_1 = -\frac{1}{4} - \left(-\frac{5}{8}\right) = -\frac{1}{4} + \frac{5}{8}
\]
To subtract these fractions, we need a common denominator:
\[
- \frac{1}{4} = - \frac{2}{8}
\]
Thus,
\[
x_2 - x_1 = -\frac{2}{8} + \frac{5}{8} = \frac{3}{8}
\]

### Step 2: Subtract the \(y\)-coordinates:
\[
y_2 - y_1 = -\frac{5}{8} - \left(-\frac{2}{5}\right) = -\frac{5}{8} + \frac{2}{5}
\]
To subtract these fractions, we find a common denominator (which is 40):
\[
- \frac{5}{8} = - \frac{25}{40}, \quad \frac{2}{5} = \frac{16}{40}
\]
Thus,
\[
y_2 - y_1 = - \frac{25}{40} + \frac{16}{40} = - \frac{9}{40}
\]

### Step 3: Plug into the distance formula:
\[
d = \sqrt{\left(\frac{3}{8}\right)^2 + \left(-\frac{9}{40}\right)^2}
\]
\[
d = \sqrt{\frac{9}{64} + \frac{81}{1600}}
\]

### Step 4: Find a common denominator and add the fractions:
Convert \(\frac{9}{64}\) to a fraction with a denominator of 1600:
\[
\frac{9}{64} = \frac{225}{1600}
\]
Now add:
\[
d = \sqrt{\frac{225}{1600} + \frac{81}{1600}} = \sqrt{\frac{306}{1600}}
\]

### Step 5: Simplify the square root:
\[
d = \frac{\sqrt{306}}{40}
\]

The distance between the points is approximately:
\[
d \approx \frac{17.49}{40} \approx 0.437
\]

Would you like more details on any step?

Here are five related questions to explore:
1. What is the distance formula and how is it derived?
2. How do you simplify square roots with fractions?
3. How do you find the midpoint between two points?
4. What is the significance of using the Euclidean distance in geometry?
5. How do you apply the distance formula in 3D space?

**Tip:** When adding or subtracting fractions, always find a common denominator to ensure accurate results.

Learn how to calculate the distance between the points (-5/8, -2/5) and (-1/4, -5/8) using the distance formula. This step-by-step guide covers fraction subtraction, the Pythagorean Theorem, and simplification techniques. Suitable for students in Grades 8-10.

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