Given the image:

We have points $A = (3, 7)$ and $B = (5, 5.5)$ on a straight line $ABC$, and the ratio of $AB : BC = 5 : 2$. The goal is to determine the coordinates of point $C$ along the line such that the segment lengths $AB$ and $BC$ maintain this ratio.

Step-by-Step Solution

1. Find the Length of Segment $AB$: $AB = \sqrt{(5 - 3)^2 + (5.5 - 7)^2} = \sqrt{2^2 + (-1.5)^2} = \sqrt{4 + 2.25} = \sqrt{6.25} = 2.5$

2. Determine the Total Length of $AC$ Using the Ratio $AB : BC = 5 : 2$: Since $AB : BC = 5 : 2$, let $AB = 5x$ and $BC = 2x$. $5x = 2.5 \Rightarrow x = \frac{2.5}{5} = 0.5$ Therefore, $BC = 2x = 2 \times 0.5 = 1$.

   So, the length $AC = AB + BC = 2.5 + 1 = 3.5$.

3. Calculate the Direction Vector from $A$ to $B$: The direction vector $\vec{AB}$ is: $\vec{AB} = (5 - 3, 5.5 - 7) = (2, -1.5)$

4. Scale the Direction Vector to Find $C$: We need to extend from $A$ to $C$ by a total of $3.5$ units. Since $AB$ is 2.5 units, the direction remains the same. We find $C$ by moving $3.5$ units along the line.

   The unit vector in the direction from $A$ to $B$ is: $\text{Unit Vector} = \left( \frac{2}{2.5}, \frac{-1.5}{2.5} \right) = \left( 0.8, -0.6 \right)$

   Now, multiply this unit vector by $3.5$ and add it to $A$: $C = A + 3.5 \times (0.8, -0.6) = (3, 7) + (2.8, -2.1) = (5.8, 4.9)$

Conclusion

The coordinates of point $C$ are $\mathbf{(5.8, 4.9)}$.

Would you like further details or have any questions?

Related Questions:

1. How would you find point $C$ if the ratio were different, say $AB : BC = 3 : 1$?
2. What would be the coordinates of $C$ if $B$ were located elsewhere on the line?
3. How can you use vector notation to solve similar problems more generally?
4. How does the concept of section formulas apply in this context?
5. What other applications exist for dividing lines in a given ratio?

Tip:

When working with ratios on a line, always calculate direction vectors and scale them according to the given ratio for accurate results.