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!