O (0, 0), A (3, 5) and B (-5, -3) are the vertices of triangle OAB. Find :
the equation of median of triangle OAB through vertex O. 

Answer

To find the equation of the median of triangle OAB through vertex O, we will follow these steps:

### Step 1: Identify the coordinates of the vertices
The vertices of triangle OAB are given as:
- O (0, 0)
- A (3, 5)
- B (-5, -3)

### Step 2: Find the midpoint D of side AB
To find the midpoint D of the line segment AB, we use the midpoint formula:
\[ D\left(\frac{x_1 + x_2}{2}, \frac{y_1 + y_2}{2}\right) \]
where \(A(x_1, y_1) = (3, 5)\) and \(B(x_2, y_2) = (-5, -3)\).

Calculating the coordinates of D:
\[ D\left(\frac{3 + (-5)}{2}, \frac{5 + (-3)}{2}\right) = D\left(\frac{-2}{2}, \frac{2}{2}\right) = D(-1, 1) \]

### Step 3: Find the slope of line OD
To find the slope of line OD, we use the slope formula:
\[ m = \frac{y_2 - y_1}{x_2 - x_1} \]
Here, we take \(O(0, 0)\) as \((x_1, y_1)\) and \(D(-1, 1)\) as \((x_2, y_2)\).

Calculating the slope:
\[ m = \frac{1 - 0}{-1 - 0} = \frac{1}{-1} = -1 \]

### Step 4: Write the equation of line OD
Using the point-slope form of the equation of a line:
\[ y - y_1 = m(x - x_1) \]
Substituting \(O(0, 0)\) and the slope \(m = -1\):
\[ y - 0 = -1(x - 0) \]
This simplifies to:
\[ y = -x \]

### Step 5: Rearranging the equation
We can rearrange the equation to standard form:
\[ y + x = 0 \]

### Final Answer
The equation of the median of triangle OAB through vertex O is:
\[ y + x = 0 \]

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