The points (1,5) , (2,4) and (3,3) are

Options

A. vertices of an equilateral triangle
B. vertices of an isosceles triangle
C. vertices of a right-angle triangle
D. collinear

Answer

To determine the relationship between the points (1,5), (2,4), and (3,3), we will check if they are collinear by calculating the slopes of the line segments formed by these points.

### Step-by-Step Solution:

1. **Identify the Points**:
Let the points be A(1, 5), B(2, 4), and C(3, 3).

2. **Calculate the Slope of Line AB**:
The formula for the slope (m) between two points (x1, y1) and (x2, y2) is given by:
\[ m = \frac{y2 - y1}{x2 - x1} \]
For points A(1, 5) and B(2, 4):
\[ m_{AB} = \frac{4 - 5}{2 - 1} = \frac{-1}{1} = -1 \]

3. **Calculate the Slope of Line BC**:
Now, we calculate the slope between points B(2, 4) and C(3, 3):
\[ m_{BC} = \frac{3 - 4}{3 - 2} = \frac{-1}{1} = -1 \]

4. **Compare the Slopes**:
We found that:
\[ m_{AB} = -1 \quad \text{and} \quad m_{BC} = -1 \]
Since the slopes of AB and BC are equal, it indicates that the points A, B, and C are collinear.

5. **Conclusion**:
Therefore, the points (1,5), (2,4), and (3,3) are collinear.