The middle point of A (1, 2) and B (x, y) is C (2, 4). If BD is perpendicular to AB such that CD = 3 unit, then what is the length BD ?

Options

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

Answer

To solve the problem step by step, we will follow these steps:

### Step 1: Find the coordinates of point B
We know that point C is the midpoint of points A (1, 2) and B (x, y). The formula for the midpoint C is given by:

\[ C\left(\frac{x_1 + x_2}{2}, \frac{y_1 + y_2}{2}\right) \]

Substituting the coordinates of A and C into the formula:

\[ C\left(2, 4\right) = \left(\frac{1 + x}{2}, \frac{2 + y}{2}\right) \]

This gives us two equations:

1. \(\frac{1 + x}{2} = 2\)
2. \(\frac{2 + y}{2} = 4\)

### Step 2: Solve for x
From the first equation:

\[ \frac{1 + x}{2} = 2 \]

Multiplying both sides by 2:

\[ 1 + x = 4 \]

Subtracting 1 from both sides:

\[ x = 3 \]

### Step 3: Solve for y
From the second equation:

\[ \frac{2 + y}{2} = 4 \]

Multiplying both sides by 2:

\[ 2 + y = 8 \]

Subtracting 2 from both sides:

\[ y = 6 \]

Thus, the coordinates of point B are (3, 6).

### Step 4: Find the length of BC
Now that we have the coordinates of B (3, 6) and C (2, 4), we can find the distance BC using the distance formula:

\[ BC = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2} \]

Substituting the coordinates of B and C:

\[ BC = \sqrt{(3 - 2)^2 + (6 - 4)^2} \]

Calculating:

\[ BC = \sqrt{(1)^2 + (2)^2} = \sqrt{1 + 4} = \sqrt{5} \]

### Step 5: Use Pythagorean theorem to find BD
We know that BD is perpendicular to AB, and CD = 3 units. We can apply the Pythagorean theorem in triangle BCD:

\[ CD^2 = BD^2 + BC^2 \]

Substituting the known values:

\[ 3^2 = BD^2 + (\sqrt{5})^2 \]

Calculating:

\[ 9 = BD^2 + 5 \]

### Step 6: Solve for BD
Rearranging the equation:

\[ BD^2 = 9 - 5 \]
\[ BD^2 = 4 \]

Taking the square root:

\[ BD = \sqrt{4} = 2 \]

### Final Answer
The length of BD is **2 units**.

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