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.