If the vertices of triangle ABC are (0,0) , (5,1) and (7,3) respectively, then find the length of the median through A

Options

A. `5`
B. ` sqrt""27`
C. `sqrt""40`
D. None of these

Answer

To find the length of the median through vertex A of triangle ABC with vertices A(0,0), B(5,1), and C(7,3), we can follow these steps:

### Step 1: Identify the Coordinates of the Vertices
The vertices of the triangle are given as:
- A(0, 0)
- B(5, 1)
- C(7, 3)

### Step 2: Find the Midpoint D of Side BC
To find the length of the median through A, we first need to find the midpoint D of side BC. The formula for the midpoint D(x, y) between two points B(x1, y1) and C(x2, y2) is:
\[ D_x = \frac{x_1 + x_2}{2}, \quad D_y = \frac{y_1 + y_2}{2} \]

Substituting the coordinates of B and C:
\[ D_x = \frac{5 + 7}{2} = \frac{12}{2} = 6 \]
\[ D_y = \frac{1 + 3}{2} = \frac{4}{2} = 2 \]
Thus, the coordinates of D are D(6, 2).

### Step 3: Calculate the Length of Median AD
Now, we will calculate the length of the median AD using the distance formula. The distance \(d\) between two points A(x1, y1) and D(x2, y2) is given by:
\[ d = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2} \]

Substituting the coordinates of A(0, 0) and D(6, 2):
\[ AD = \sqrt{(6 - 0)^2 + (2 - 0)^2} \]
\[ AD = \sqrt{6^2 + 2^2} = \sqrt{36 + 4} = \sqrt{40} \]

### Step 4: Final Result
The length of the median AD is \(\sqrt{40}\).