Using distance formula, show that the points `A(1,-1),B(5,2)andC(9,5)` are collinear.

Answer

To show that the points A(1, -1), B(5, 2), and C(9, 5) are collinear using the distance formula, we will follow these steps:

### Step 1: Calculate the distance between points A and C
We will use the distance formula:
\[ d = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2} \]
For points A(1, -1) and C(9, 5):
- \(x_1 = 1\), \(y_1 = -1\)
- \(x_2 = 9\), \(y_2 = 5\)

Substituting these values into the distance formula:
\[ AC = \sqrt{(9 - 1)^2 + (5 - (-1))^2} \]
Calculating inside the square root:
\[ = \sqrt{(8)^2 + (6)^2} = \sqrt{64 + 36} = \sqrt{100} = 10 \]

### Step 2: Calculate the distance between points A and B
Using the same distance formula for points A(1, -1) and B(5, 2):
- \(x_1 = 1\), \(y_1 = -1\)
- \(x_2 = 5\), \(y_2 = 2\)

Substituting these values:
\[ AB = \sqrt{(5 - 1)^2 + (2 - (-1))^2} \]
Calculating:
\[ = \sqrt{(4)^2 + (3)^2} = \sqrt{16 + 9} = \sqrt{25} = 5 \]

### Step 3: Calculate the distance between points B and C
Now, we calculate the distance between points B(5, 2) and C(9, 5):
- \(x_1 = 5\), \(y_1 = 2\)
- \(x_2 = 9\), \(y_2 = 5\)

Substituting these values:
\[ BC = \sqrt{(9 - 5)^2 + (5 - 2)^2} \]
Calculating:
\[ = \sqrt{(4)^2 + (3)^2} = \sqrt{16 + 9} = \sqrt{25} = 5 \]

### Step 4: Check the collinearity condition
For points A, B, and C to be collinear, the distance AC should equal the sum of distances AB and BC:
\[ AC = AB + BC \]
Substituting the distances we calculated:
\[ 10 = 5 + 5 \]
This is true.

### Conclusion
Since \(AC = AB + BC\) holds true, we conclude that the points A(1, -1), B(5, 2), and C(9, 5) are collinear.

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