https://file.aiquickdraw.com/mathbot/file/1726188896305wmdaek4l.jpg

We are given two points on a Cartesian plane: 

- Point \( A(4, 2) \)
- Point \( B(22, 12) \)

We need to find the coordinates of point \( C \), which is located \( \frac{3}{5} \) of the way along segment \( AB \), starting from point \( A \).

The formula to find a point that divides a line segment in a given ratio is:

\[
C(x, y) = \left( x_1 + \frac{m}{m+n} \times (x_2 - x_1),\ y_1 + \frac{m}{m+n} \times (y_2 - y_1) \right)
\]

Here:
- \( A(x_1, y_1) = (4, 2) \)
- \( B(x_2, y_2) = (22, 12) \)
- \( \frac{m}{m+n} = \frac{3}{5} \)

Now applying the formula for both the \( x \)-coordinate and \( y \)-coordinate of point \( C \):

1. **For the \( x \)-coordinate**:
\[
x_C = 4 + \frac{3}{5} \times (22 - 4) = 4 + \frac{3}{5} \times 18 = 4 + \frac{54}{5} = \frac{20}{5} + \frac{54}{5} = \frac{74}{5}
\]

2. **For the \( y \)-coordinate**:
\[
y_C = 2 + \frac{3}{5} \times (12 - 2) = 2 + \frac{3}{5} \times 10 = 2 + 6 = 8
\]

Thus, the coordinates of point \( C \) are \( \left( \frac{74}{5}, 8 \right) \).

This matches option **A** in the provided choices.

---

Would you like more details or explanations on this solution? 

Here are 5 related questions to explore:

1. How do you derive the section formula?
2. What are some real-world applications of dividing a line segment in a given ratio?
3. How would the formula change if point \( C \) were located outside segment \( AB \)?
4. Can you derive the midpoint formula from this general ratio division formula?
5. How do changes in the ratio \( \frac{m}{m+n} \) affect the location of point \( C \)?

**Tip**: Always verify your solution by plugging values back into the problem, especially when working with fractions!

A third object must be placed on segment AB, at a point C located 3/5 of the way along segment AB, starting from point A. Identify the coordinates of point C.

Learn how to determine the coordinates of point C, located 3/5 of the way along segment AB. This solution covers key concepts of coordinate geometry and uses the section formula for ratio division. Suitable for students in Grades 9-12.

Math Problem Statement

Solution

Ask a new question for Free

By Image

Math Problem Analysis

Mathematical Concepts

Formulas

Theorems

Suitable Grade Level

Related Recommendation