Plot the points `A (1, 1), B (4, 7) and C (4, 10)` on a graph paper. Connect A and B, and also A and C.
  Which segment appears to have the steeper slope, AB or AC ?
Justify your conclusion by calculating the slopes of AB and AC. 

Answer

To solve the problem, we will follow these steps:

### Step 1: Plot the points A, B, and C on the graph paper.
- **Point A (1, 1)**: Start at the origin (0, 0), move 1 unit to the right (x-axis) and 1 unit up (y-axis).
- **Point B (4, 7)**: From the origin, move 4 units to the right and 7 units up.
- **Point C (4, 10)**: From the origin, move 4 units to the right and 10 units up.

### Step 2: Connect the points A and B, and A and C.
- Draw a straight line from point A to point B.
- Draw another straight line from point A to point C.

### Step 3: Determine which segment appears to have a steeper slope.
- Visually inspect the lines AB and AC on the graph. The line that rises more steeply from left to right has the steeper slope.

### Step 4: Calculate the slopes of segments AB and AC.
- **Slope formula**: The slope (m) between two points (x1, y1) and (x2, y2) is given by:
\[ m = \frac{y2 - y1}{x2 - x1} \]

#### Calculate the slope of segment AB:
- Coordinates of A: (1, 1) and B: (4, 7)
- Using the slope formula:
\[ m_{AB} = \frac{7 - 1}{4 - 1} = \frac{6}{3} = 2 \]

#### Calculate the slope of segment AC:
- Coordinates of A: (1, 1) and C: (4, 10)
- Using the slope formula:
\[ m_{AC} = \frac{10 - 1}{4 - 1} = \frac{9}{3} = 3 \]

### Step 5: Compare the slopes.
- Slope of AB: 2
- Slope of AC: 3
- Since 3 > 2, the segment AC has a steeper slope than segment AB.

### Conclusion:
- The segment AC appears to have a steeper slope than segment AB, as confirmed by our calculations.

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