determine the coordinate of a point 1/4 the distance from (-8,10) to (4,-6)

To determine the coordinates of a point that is \(\frac{1}{4}\) of the distance from \((-8, 10)\) to \((4, -6)\), we can use the section formula for dividing a line segment internally.

### Step 1: Identify the coordinates
Let \( A = (-8, 10) \) and \( B = (4, -6) \).

### Step 2: Use the section formula
The section formula for dividing the line segment joining two points \( A(x_1, y_1) \) and \( B(x_2, y_2) \) internally in the ratio \( m:n \) is:

\[
(x, y) = \left( \frac{mx_2 + nx_1}{m+n}, \frac{my_2 + ny_1}{m+n} \right)
\]

For this problem, the ratio \( m:n = 1:3 \) because the point is \(\frac{1}{4}\) of the distance from \( A \) to \( B \).

### Step 3: Substitute the values into the formula
\[
x = \frac{1 \cdot 4 + 3 \cdot (-8)}{1+3} = \frac{4 - 24}{4} = \frac{-20}{4} = -5
\]
\[
y = \frac{1 \cdot (-6) + 3 \cdot 10}{1+3} = \frac{-6 + 30}{4} = \frac{24}{4} = 6
\]

### Step 4: Write the final coordinates
The coordinates of the point that is \(\frac{1}{4}\) of the distance from \((-8, 10)\) to \((4, -6)\) are \((-5, 6)\).

Would you like further details or have any other questions?

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Here are 8 related questions:
1. How do you determine the midpoint of a line segment?
2. What is the formula for dividing a line segment externally in a given ratio?
3. How do you find the distance between two points in a coordinate plane?
4. Can you explain the concept of the centroid of a triangle?
5. What is the equation of a line passing through two points?
6. How would you determine the area of a triangle given its vertices?
7. What is the section formula for three-dimensional space?
8. How can you find the slope of a line given two points?

**Tip:** Always double-check the ratio used in the section formula to ensure you are calculating the correct point along the segment.

Learn how to find the coordinates of a point that is 1/4 of the distance from (-8, 10) to (4, -6). This problem involves using the section formula to divide a line segment internally in a given ratio. Suitable for high school students studying coordinate geometry.

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