Properties of Triangle Medians in Coordinate Geometry

Hi Melek, let's look at this coordinate geometry problem exploring the properties of medians in a triangle.

First, let's identify our vertices from the diagram. Vertex O is at the origin, A is at two a, zero, and B is at two b, two c.

For part a, we need the equation for median O P. P is the midpoint of side A B.

Substituting the coordinates of A and B, we find that P is at a plus b, c.

Since median O P passes through the origin 0, 0 and P, its gradient m is c over a plus b.

Using the line equation y equals m x, we get y equals c over a plus b times x.

Cross multiplying and rearranging, we get c x minus, bracket, a plus b, bracket, y equals zero. This proves part a.

Now for part b, we find median A Q. Q is the midpoint of side O B.

The gradient of A Q, from A at two a, zero to Q at b, c, is c minus zero over b minus two a.

Using the point-slope form with point A, we have y minus zero equals the gradient times x minus two a.

Multiply both sides by b minus two a to clear the fraction.

Expanding the right side gives c x minus two a c. Moving the terms, we get c x minus, bracket, b minus two a, bracket, y equals two a c. That's our second equation.