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

Answer

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

### Step 1: Understand the Condition for Collinearity
Three points A, B, and C are collinear if the area of the triangle formed by these points is zero. The formula for the area of a triangle given three points (x1, y1), (x2, y2), and (x3, y3) is:

\[ \text{Area} = \frac{1}{2} \left| x_1(y_2 - y_3) + x_2(y_3 - y_1) + x_3(y_1 - y_2) \right| \]

### Step 2: Substitute the Coordinates
Substituting the coordinates of points A(1, 2), B(2, 4), and C(3, a) into the area formula:

\[ \text{Area} = \frac{1}{2} \left| 1(4 - a) + 2(a - 2) + 3(2 - 4) \right| = 0 \]

### Step 3: Simplify the Expression
Now, simplify the expression inside the absolute value:

\[ = \frac{1}{2} \left| 4 - a + 2a - 4 + 3(2 - 4) \right| \]
\[ = \frac{1}{2} \left| 4 - a + 2a - 4 + 6 - 12 \right| \]
\[ = \frac{1}{2} \left| a - 6 \right| = 0 \]

### Step 4: Solve for 'a'
Since the area is zero, we set the expression inside the absolute value to zero:

\[ |a - 6| = 0 \]
This implies:
\[ a - 6 = 0 \quad \Rightarrow \quad a = 6 \]

### Step 5: Find the Coordinates of Point C
Now that we have the value of 'a', the coordinates of point C are:
\[ C(3, 6) \]

### Step 6: Calculate the Length of BC
To find the length of segment BC, we use the distance formula:

\[ \text{Distance} = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2} \]
Where B(2, 4) and C(3, 6):
\[ BC = \sqrt{(3 - 2)^2 + (6 - 4)^2} \]
\[ = \sqrt{1^2 + 2^2} \]
\[ = \sqrt{1 + 4} = \sqrt{5} \]

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

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