To find the coordinates of the point that is $\frac{1}{4}$ of the way from point $A(-6, -3)$ to point $B(10, 9)$, we can use the formula for the coordinates of a point dividing a line segment in a given ratio.

The formula is:

$P(x, y) = \left( x_1 + \frac{m}{m+n}(x_2 - x_1), \ y_1 + \frac{m}{m+n}(y_2 - y_1) \right)$

Where:

• $A(x_1, y_1) = (-6, -3)$
• $B(x_2, y_2) = (10, 9)$
• The ratio is $1:3$, because $\frac{1}{4}$ means the point is $1$ part of the way from $A$ to $B$, and $3$ parts away from $B$.

Now, apply this formula:

$P(x, y) = \left( -6 + \frac{1}{4}(10 + 6), \ -3 + \frac{1}{4}(9 + 3) \right)$

Simplifying the expression:

$P(x, y) = \left( -6 + \frac{1}{4}(16), \ -3 + \frac{1}{4}(12) \right)$ $P(x, y) = \left( -6 + 4, \ -3 + 3 \right)$ $P(x, y) = (-2, 0)$

So, the coordinates of the point $\frac{1}{4}$ of the way from $A$ to $B$ are $(-2, 0)$.

Would you like more details or further explanation?

Here are some related questions:

1. What is the midpoint between two points?
2. How do you find the point dividing a segment in any given ratio?
3. How do you calculate the distance between two points in a plane?
4. What is the section formula for internal division?
5. How do you handle ratio problems when the point divides externally?

Tip: Always remember that the section formula changes for external division, using a negative sign for the ratio when the point lies outside the segment.