The vertices of a triangle are A (3,-2), B (-2, 1) and C (5,2). Then the length of the median through B is 

Options

A. `sqrt(67)` units
B. `sqrt(37)` units
C. `sqrt(35)` units
D. 6 units

Answer

To find the length of the median through vertex B of triangle ABC with vertices A(3, -2), B(-2, 1), and C(5, 2), we will follow these steps:

### Step 1: Find the midpoint of side AC
The median from vertex B will bisect the opposite side AC. To find the midpoint M of side AC, we use the midpoint formula:
\[ M = \left( \frac{x_1 + x_2}{2}, \frac{y_1 + y_2}{2} \right) \]
where \( A(3, -2) \) and \( C(5, 2) \).

Calculating the coordinates of M:
\[ M_x = \frac{3 + 5}{2} = \frac{8}{2} = 4 \]
\[ M_y = \frac{-2 + 2}{2} = \frac{0}{2} = 0 \]
Thus, the midpoint M is \( (4, 0) \).

### Step 2: Use the distance formula to find the length of median BM
Now we need to find the length of the median BM, where \( B(-2, 1) \) and \( M(4, 0) \). We will use the distance formula:
\[ d = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2} \]
Substituting the coordinates of B and M:
\[ d = \sqrt{(4 - (-2))^2 + (0 - 1)^2} \]
Calculating the differences:
\[ d = \sqrt{(4 + 2)^2 + (0 - 1)^2} = \sqrt{(6)^2 + (-1)^2} \]
Calculating the squares:
\[ d = \sqrt{36 + 1} = \sqrt{37} \]

### Final Answer
The length of the median through B is \( \sqrt{37} \).

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