If the points A(1, 2), B(2, 4) and C(3, a) are collinear, what is the length BC ?

Options

A. `sqrt(2)` unit
B. `sqrt(3)` unit
C. `sqrt(5)` unit
D. 5 unit

Explanation

To find the length of segment BC given that points A(1, 2), B(2, 4), and C(3, a) are collinear, we can follow these steps:

### Step 1: Find the slope of line AB
The slope (m) of a line through two points (x1, y1) and (x2, y2) is given by the formula:
\[ m = \frac{y2 - y1}{x2 - x1} \]
For points A(1, 2) and B(2, 4):
- \(x_1 = 1\), \(y_1 = 2\)
- \(x_2 = 2\), \(y_2 = 4\)

Calculating the slope:
\[ m_{AB} = \frac{4 - 2}{2 - 1} = \frac{2}{1} = 2 \]

### Step 2: Use the slope to find the value of 'a' for point C
Since points A, B, and C are collinear, the slope of line BC must also be equal to the slope of line AB. The slope of line BC can be calculated as:
\[ m_{BC} = \frac{a - 4}{3 - 2} \]
Setting the slopes equal:
\[ m_{AB} = m_{BC} \]
\[ 2 = a - 4 \]
Solving for 'a':
\[ a = 2 + 4 = 6 \]

### Step 3: Calculate the length of segment BC
Now that we have the coordinates of point C as (3, 6), we can find the length of segment BC using the distance formula:
\[ BC = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2} \]
For points B(2, 4) and C(3, 6):
- \(x_1 = 2\), \(y_1 = 4\)
- \(x_2 = 3\), \(y_2 = 6\)

Calculating the distance:
\[ BC = \sqrt{(3 - 2)^2 + (6 - 4)^2} = \sqrt{(1)^2 + (2)^2} = \sqrt{1 + 4} = \sqrt{5} \]

### Final Answer
The length of segment BC is \(\sqrt{5}\) units.

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