https://file.aiquickdraw.com/mathbot/file/1726576885053n7dtg03e.jpg

In the image, the question asks about a line passing through two points, \( A(1, 4) \) and \( B(3, -2) \), and seeks to determine which of the labeled points (I, II, III, IV, V) on the graph is closest to this line.

### Step 1: Equation of the Line
To find the equation of the line passing through points \( A(1, 4) \) and \( B(3, -2) \), we first calculate the slope \( m \):

\[
m = \frac{y_2 - y_1}{x_2 - x_1} = \frac{-2 - 4}{3 - 1} = \frac{-6}{2} = -3
\]

Now, using the point-slope form of the line equation \( y - y_1 = m(x - x_1) \), and taking point \( A(1, 4) \):

\[
y - 4 = -3(x - 1)
\]

Expanding and solving for \( y \):

\[
y - 4 = -3x + 3
\]
\[
y = -3x + 7
\]

So, the equation of the line is:

\[
y = -3x + 7
\]

### Step 2: Check Distance of Labeled Points to the Line
We need to check which of the labeled points is closest to this line. Let's check their coordinates and use the formula for the perpendicular distance from a point \( (x_1, y_1) \) to a line \( Ax + By + C = 0 \):

\[
\text{Distance} = \frac{|Ax_1 + By_1 + C|}{\sqrt{A^2 + B^2}}
\]

Where the line is in the form \( -3x + y - 7 = 0 \), so \( A = -3 \), \( B = 1 \), and \( C = -7 \).

We can now compute the distance for each point labeled I, II, III, IV, and V. Would you like me to compute these distances or clarify anything further?

Here are some follow-up questions to expand on this problem:

1. How do we calculate the perpendicular distance from a point to a line?
2. What is the geometric interpretation of the slope in the equation of a line?
3. Can we derive the equation of a line given any two points using a different method?
4. How does changing the slope affect the steepness of a line on a graph?
5. What are the real-world applications of finding the shortest distance from a point to a line?

**Tip:** When working with line equations, always double-check the slope and point substitutions to avoid common algebraic errors.

Koordinat düzlem grafiğinde A(1, 4) ve B(3, -2) noktalarından geçen bir doğru bulunmaktadır. Buna göre grafikte verilen noktalardan hangisi bu doğruya daha yakındır?

Solve a coordinate geometry problem by finding the equation of a line passing through two points and determining which given point is closest to the line using the perpendicular distance formula. Suitable for students in Grades 9-11.

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