We are given two points, $A(85, 0)$ and $B(-70, 90)$, and we need to find the coordinates of a point $C(x, y)$ that is two-fifths of the way from $A$ to $B$.

Formula for finding a point on a line segment

The coordinates of a point $C$ that divides the line segment joining points $A(x_1, y_1)$ and $B(x_2, y_2)$ in a ratio $m:n$ can be calculated using the section formula: $C(x, y) = \left( \frac{m \cdot x_2 + n \cdot x_1}{m + n}, \frac{m \cdot y_2 + n \cdot y_1}{m + n} \right)$ In this case, we want $C$ to be two-fifths of the distance from $A$ to $B$. So, the ratio $m:n$ is 2:3 (since two-fifths of the total distance leaves three-fifths).

Given points:

• $A(85, 0)$
• $B(-70, 90)$
• Ratio: $m:n = 2:3$

Let's calculate the coordinates of $C$.The coordinates of point $C$, which is two-fifths of the way from $A(85, 0)$ to $B(-70, 90)$, are:

$C(x, y) = (23.0, 36.0)$

Would you like further details on the steps or clarification on any part of the process?

Here are 5 related questions to further explore this concept:

1. How do you find a point that divides a segment in any other given ratio?
2. What is the formula for finding the midpoint of a segment?
3. How would you calculate the distance between two points in a plane?
4. What would be the coordinates of the point three-fifths of the way from $A$ to $B$?
5. How is the section formula related to vector interpolation?

Tip: The section formula works for dividing a line segment internally or externally depending on the ratio you use!