If `A(-2,4) , B(0,0)` and C(4,2) are the vertices of `DeltaABC` , find the length of the median through A .

Answer

To find the length of the median through vertex A of triangle ABC with vertices A(-2, 4), B(0, 0), and C(4, 2), we can follow these steps:

### Step 1: Find the midpoint of line segment BC
The median through A will intersect line segment BC at its midpoint, which we will denote as point E. The coordinates of point E can be calculated using the midpoint formula:

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

where \(B(0, 0)\) and \(C(4, 2)\).

### Calculation:
\[ E\left(\frac{0 + 4}{2}, \frac{0 + 2}{2}\right) = E\left(\frac{4}{2}, \frac{2}{2}\right) = E(2, 1) \]

### Step 2: Use the distance formula to find the length of median AE
Now that we have the coordinates of points A and E, we can find the length of median AE using the distance formula:

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

where \(A(-2, 4)\) and \(E(2, 1)\).

### Calculation:
\[ d = \sqrt{(2 - (-2))^2 + (1 - 4)^2} \]
\[ = \sqrt{(2 + 2)^2 + (1 - 4)^2} \]
\[ = \sqrt{(4)^2 + (-3)^2} \]
\[ = \sqrt{16 + 9} \]
\[ = \sqrt{25} \]
\[ = 5 \]

### Conclusion:
The length of the median through vertex A is 5 units.

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