Using slopes find out which of the sets of three points lie on a straight line.

a) A (-1, 2), B (2, 1), C (5,0) b) A (-10, 2), B (1,-2), C (6,5)

To determine if three points lie on a straight line, we can calculate the slopes between pairs of points. If the slopes between all pairs are the same, the points lie on the same straight line.

### Part a: \( A(-1, 2), B(2, 1), C(5, 0) \)

1. **Slope between A and B**:
   \[
   m_{AB} = \frac{y_B - y_A}{x_B - x_A} = \frac{1 - 2}{2 - (-1)} = \frac{-1}{3} = -\frac{1}{3}
   \]

2. **Slope between B and C**:
   \[
   m_{BC} = \frac{y_C - y_B}{x_C - x_B} = \frac{0 - 1}{5 - 2} = \frac{-1}{3} = -\frac{1}{3}
   \]

3. **Slope between A and C**:
   \[
   m_{AC} = \frac{y_C - y_A}{x_C - x_A} = \frac{0 - 2}{5 - (-1)} = \frac{-2}{6} = -\frac{1}{3}
   \]

Since \( m_{AB} = m_{BC} = m_{AC} \), the points \( A(-1, 2), B(2, 1), C(5, 0) \) lie on the same straight line.

### Part b: \( A(-10, 2), B(1, -2), C(6, 5) \)

1. **Slope between A and B**:
   \[
   m_{AB} = \frac{y_B - y_A}{x_B - x_A} = \frac{-2 - 2}{1 - (-10)} = \frac{-4}{11} = -\frac{4}{11}
   \]

2. **Slope between B and C**:
   \[
   m_{BC} = \frac{y_C - y_B}{x_C - x_B} = \frac{5 - (-2)}{6 - 1} = \frac{7}{5}
   \]

3. **Slope between A and C**:
   \[
   m_{AC} = \frac{y_C - y_A}{x_C - x_A} = \frac{5 - 2}{6 - (-10)} = \frac{3}{16}
   \]

Since \( m_{AB} \neq m_{BC} \neq m_{AC} \), the points \( A(-10, 2), B(1, -2), C(6, 5) \) do **not** lie on the same straight line.

### Conclusion:
- **Set a)**: The points lie on the same line.
- **Set b)**: The points do not lie on the same line.

Would you like further details or have any questions?

Here are 5 related questions:
1. How do you find the equation of a line using two points?
2. What is the significance of the slope being constant between points?
3. Can you check if points lie on a line by using the equation of a line?
4. How do you find the midpoint between two points on a coordinate plane?
5. What are the properties of a line with a slope of zero?

**Tip**: If three points are collinear, the area of the triangle formed by them will be zero.

Using slopes find out which of the sets of three points lie on a straight line. a) A (-1, 2), B (2, 1), C (5,0) b) A (-10, 2), B (1,-2), C (6,5)

Learn how to determine if the sets of points A(-1, 2), B(2, 1), C(5, 0) and A(-10, 2), B(1, -2), C(6, 5) lie on the same straight line by calculating slopes. This step-by-step solution uses the slope formula and explores key geometric concepts. Suitable for students in Grades 8-10.

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