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 whether a set of three points lies on a straight line, we can use the concept of slopes. The points will be collinear if the slope between any two pairs of points is the same.

### Formula for slope:
\[
\text{slope (m)} = \frac{y_2 - y_1}{x_2 - x_1}
\]

We'll calculate the slopes for each set of points, comparing the slopes between:
1. \( A \) and \( B \)
2. \( B \) and \( C \)
3. \( A \) and \( C \)

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

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

2. **Slope between \( B(2, 1) \) and \( C(5, 0):**
\[
m_{BC} = \frac{0 - 1}{5 - 2} = \frac{-1}{3} = -\frac{1}{3}
\]

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

Since \( m_{AB} = m_{BC} = m_{AC} = -\frac{1}{3} \), the points \( A(-1, 2) \), \( B(2, 1) \), and \( C(5, 0) \) are **collinear**.

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

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

2. **Slope between \( B(1, -2) \) and \( C(6, -5):**
\[
m_{BC} = \frac{-5 - (-2)}{6 - 1} = \frac{-3}{5} = -\frac{3}{5}
\]

3. **Slope between \( A(-10, 2) \) and \( C(6, -5):**
\[
m_{AC} = \frac{-5 - 2}{6 - (-10)} = \frac{-7}{16}
\]

Since \( m_{AB} \neq m_{BC} \neq m_{AC} \), the points \( A(-10, 2) \), \( B(1, -2) \), and \( C(6, -5) \) are **not collinear**.

### Conclusion:
- Set a) \( A(-1, 2) \), \( B(2, 1) \), and \( C(5, 0) \) are collinear.
- Set b) \( A(-10, 2) \), \( B(1, -2) \), and \( C(6, -5) \) are not collinear.

Would you like further details or have any other questions?

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**Relative questions:**
1. What is the general condition for three points to be collinear?
2. How can the slope formula be derived geometrically?
3. How do parallel lines relate to slopes?
4. How does changing the position of one point affect collinearity?
5. Can three-dimensional points be checked for collinearity similarly?

**Tip:** When checking for collinearity, ensuring the slopes between different pairs of points are identical is crucial!

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 whether sets of points A(-1,2), B(2,1), C(5,0) and A(-10,2), B(1,-2), C(6,-5) lie on a straight line using the slope formula. This problem involves key geometry concepts, including slopes and collinearity, suitable for Grades 8-10.

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