Point A (2, -3) lies on the line segment joining the points P(-4,- 3) and Q (1, -3).

Answer

To prove that point A (2, -3) lies on the line segment joining points P (-4, -3) and Q (1, -3), we can follow these steps:

### Step 1: Determine the equation of the line segment joining points P and Q.
The formula for the equation of a line passing through two points (x1, y1) and (x2, y2) is given by:

\[ \frac{y - y_1}{x - x_1} = \frac{y_2 - y_1}{x_2 - x_1} \]

Here, we have:
- P(-4, -3) → (x1, y1) = (-4, -3)
- Q(1, -3) → (x2, y2) = (1, -3)

### Step 2: Substitute the coordinates of points P and Q into the formula.
Substituting the values into the equation, we get:

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

This simplifies to:

\[ \frac{y + 3}{x + 4} = \frac{0}{5} \]

### Step 3: Simplify the equation.
Since the right side is 0, we can conclude that:

\[ y + 3 = 0 \]

### Step 4: Solve for y.
Rearranging gives us:

\[ y = -3 \]

### Step 5: Verify if point A (2, -3) satisfies the equation.
Now we need to check if point A (2, -3) satisfies the equation \(y = -3\).

Substituting the y-coordinate of point A into the equation:

\[ y = -3 \]

Since \(y\) is indeed -3, point A satisfies the equation.

### Step 6: Conclusion.
Since point A (2, -3) satisfies the equation of the line segment joining points P and Q, we conclude that point A lies on the line segment joining P and Q.

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