Using distance formula, show that the points `(1,5), (2,4) and (3,3)` are collinear.

Answer

To show that the points \( A(1, 5) \), \( B(2, 4) \), and \( C(3, 3) \) are collinear, we will use the distance formula. The points are collinear if the sum of the distances between two pairs of points equals the distance between the third pair of points.

### Step 1: Define the points
Let:
- Point \( A = (1, 5) \)
- Point \( B = (2, 4) \)
- Point \( C = (3, 3) \)

### Step 2: Use the distance formula
The distance formula between two points \( (x_1, y_1) \) and \( (x_2, y_2) \) is given by:
\[ d = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2} \]

### Step 3: Calculate the distance \( AB \)
Using the distance formula for points \( A(1, 5) \) and \( B(2, 4) \):
\[ AB = \sqrt{(2 - 1)^2 + (4 - 5)^2} \]
\[ = \sqrt{(1)^2 + (-1)^2} \]
\[ = \sqrt{1 + 1} = \sqrt{2} \]

### Step 4: Calculate the distance \( BC \)
Now, calculate the distance between points \( B(2, 4) \) and \( C(3, 3) \):
\[ BC = \sqrt{(3 - 2)^2 + (3 - 4)^2} \]
\[ = \sqrt{(1)^2 + (-1)^2} \]
\[ = \sqrt{1 + 1} = \sqrt{2} \]

### Step 5: Calculate the distance \( AC \)
Next, calculate the distance between points \( A(1, 5) \) and \( C(3, 3) \):
\[ AC = \sqrt{(3 - 1)^2 + (3 - 5)^2} \]
\[ = \sqrt{(2)^2 + (-2)^2} \]
\[ = \sqrt{4 + 4} = \sqrt{8} = 2\sqrt{2} \]

### Step 6: Check collinearity condition
For the points to be collinear, the following condition must hold:
\[ AB + BC = AC \]
Substituting the distances we calculated:
\[ \sqrt{2} + \sqrt{2} = 2\sqrt{2} \]
\[ 2\sqrt{2} = 2\sqrt{2} \]
Since the equation holds true, we conclude that the points \( A(1, 5) \), \( B(2, 4) \), and \( C(3, 3) \) are collinear.

### Conclusion
Thus, we have shown that the points are collinear.

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