https://file.aiquickdraw.com/mathbot/file/1725418025279mqragdw5.jpg

To solve the problem in the image, we need to find the coordinates of point \( B(x, y) \) such that the ratio of \( AB \) to \( BC \) is \( 4:3 \).

### Step 1: Understand the Coordinates
Given:
- \( A(-4, -6) \)
- \( C(17, 22) \)
- Ratio \( AB : BC = 4 : 3 \)

### Step 2: Apply the Section Formula
The section formula in coordinate geometry is used to find the coordinates of a point dividing a line segment in a given ratio. If a point \( B(x, y) \) divides the line segment joining \( A(x_1, y_1) \) and \( C(x_2, y_2) \) in the ratio \( m:n \), then the coordinates of \( B \) are given by:

\[
x = \frac{mx_2 + nx_1}{m+n}
\]
\[
y = \frac{my_2 + ny_1}{m+n}
\]

Here, \( m = 4 \) and \( n = 3 \), \( A(x_1, y_1) = (-4, -6) \), \( C(x_2, y_2) = (17, 22) \).

### Step 3: Substitute the Values
\[
x = \frac{4 \cdot 17 + 3 \cdot (-4)}{4 + 3} = \frac{68 - 12}{7} = \frac{56}{7} = 8
\]
\[
y = \frac{4 \cdot 22 + 3 \cdot (-6)}{4 + 3} = \frac{88 - 18}{7} = \frac{70}{7} = 10
\]

### Final Answer
The coordinates of point \( B \) are \( (8, 10) \).

Would you like more details or have any questions?

### Related Questions:
1. How is the section formula derived in coordinate geometry?
2. What are other methods to find a point dividing a line segment?
3. Can you explain the midpoint formula as a special case of the section formula?
4. How would the coordinates change if the ratio was reversed to 3:4?
5. How can you verify the answer using the distance formula?

### Tip:
The section formula is a powerful tool not only for dividing line segments in a specific ratio but also for solving many geometric problems involving coordinate points.

The point B lies on the segment AC. Find the coordinates of B so that the ratio of AB to BC is 4 to 3. Given points A(-4, -6) and C(17, 22).

Learn how to find the coordinates of point B on segment AC where the ratio AB:BC is 4:3, using the section formula. This problem involves applying key concepts in coordinate geometry and ratios. Suitable for high school students in Grades 9-12.

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