Show that the points A(2,3) B(1,4) and c( 3,8) are colinear . Three points ABC are colinear if they lie on the same line. Proove slope of any two sides equal.

To prove that points \( A(2,3) \), \( B(1,4) \), and \( C(3,8) \) are collinear, we will compute the slopes of the line segments \( AB \) and \( AC \). If the slopes are equal, then the three points lie on the same straight line.

### Step 1: Formula for the slope
The slope of a line passing through two points \( (x_1, y_1) \) and \( (x_2, y_2) \) is given by:
\[
m = \frac{y_2 - y_1}{x_2 - x_1}
\]

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### Step 2: Slope of \( AB \)
The coordinates of \( A \) and \( B \) are \( A(2,3) \) and \( B(1,4) \).
\[
m_{AB} = \frac{4 - 3}{1 - 2} = \frac{1}{-1} = -1
\]

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### Step 3: Slope of \( AC \)
The coordinates of \( A \) and \( C \) are \( A(2,3) \) and \( C(3,8) \).
\[
m_{AC} = \frac{8 - 3}{3 - 2} = \frac{5}{1} = 5
\]

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### Step 4: Compare \( m_{AB} \) and \( m_{AC} \)
We observe that:
\[
m_{AB} = -1 \quad \text{and} \quad m_{AC} = 5
\]

Since the slopes are not equal, the points \( A \), \( B \), and \( C \) **are not collinear**.

Would you like to review this step further or solve a similar problem? Here are some questions for exploration:

1. What happens if we calculate the slope of \( BC \)?
2. How can we verify collinearity using the area of a triangle formed by three points?
3. Can you find a line equation passing through \( A \) and \( B \)? Does \( C \) satisfy the equation?
4. How do slopes help identify parallel and perpendicular lines?
5. How can we confirm if two lines intersect using their slopes?

**Tip:** If you struggle to verify collinearity, remember that three points are collinear if the area of the triangle they form is zero.

Show that the points A(2,3) B(1,4) and C(3,8) are colinear. Three points ABC are colinear if they lie on the same line. Prove slope of any two sides equal.

Learn how to prove the collinearity of points A(2,3), B(1,4), and C(3,8) by calculating the slopes of line segments AB and AC. Understand key concepts of analytical geometry and the slope formula. Suitable for students in Grades 8-10.

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