What are the coordinates of the point 14

of the way from A(–6, –3) to B(10, 9)?

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.

What are the coordinates of the point 1/4 of the way from A(–6, –3) to B(10, 9)?

Learn how to find the coordinates of the point 1/4 of the way from A(–6, –3) to B(10, 9) using the section formula. This problem covers key concepts in coordinate geometry and the section formula for dividing a line segment. Suitable for students in Grades 8-10.

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