https://file.aiquickdraw.com/mathbot/file/17267853501391uklgdie.jpg

To solve this problem, we need to find the intersection point of a line parallel to \( \overline{BC} \) and passing through point \( A \) with \( \overline{DC} \).

### Step 1: Find the slope of line \( BC \)

From the graph:

- Coordinates of \( B \): \( (0, 3) \)
- Coordinates of \( C \): \( (5, 0) \)

The slope of line \( BC \) is given by:

\[
\text{slope} = \frac{y_2 - y_1}{x_2 - x_1} = \frac{0 - 3}{5 - 0} = -\frac{3}{5}
\]

### Step 2: Find the equation of the line parallel to \( BC \) through point \( A \)

Point \( A \) is at \( (0, 1) \). Since the line passing through \( A \) is parallel to \( BC \), it has the same slope, \(-\frac{3}{5}\).

Using the point-slope form of a line, the equation is:

\[
y - y_1 = m(x - x_1)
\]

Substituting \( m = -\frac{3}{5} \), \( x_1 = 0 \), and \( y_1 = 1 \):

\[
y - 1 = -\frac{3}{5}(x - 0)
\]

Simplifying this:

\[
y = -\frac{3}{5}x + 1
\]

### Step 3: Find the intersection of the line with \( DC \)

Line \( DC \) is a vertical line at \( x = 5 \). To find the intersection, substitute \( x = 5 \) into the equation of the line:

\[
y = -\frac{3}{5}(5) + 1 = -3 + 1 = -2
\]

### Final Answer

The coordinates of the intersection of the line parallel to \( \overline{BC} \) through point \( A \) with \( \overline{DC} \) are:

\[
\boxed{(5, -2)}
\]

Would you like further details or have any questions?

### Related Questions:
1. How do you find the slope of a line given two points?
2. How do you use the point-slope form of a line equation?
3. What is the method for finding intersections of two lines?
4. How do vertical lines differ from other linear equations?
5. How do parallel lines relate to their slopes?

**Tip:** For parallel lines, the slopes are always equal.

If a line is drawn parallel to BC and through point A, at what coordinate will the intersection of this parallel line be with DC?

Learn how to find the intersection point of a line drawn parallel to BC and passing through point A, with the line DC. This geometry problem covers concepts like slope, parallel lines, and finding points of intersection in coordinate geometry. Suitable for students in Grades 9-11.

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