To find the coordinates of the third vertex of the triangle, let's break down the problem step-by-step:

Known Information:

• Two vertices of the triangle are given as:
  • $A(2, 5)$
  • $B(5, 9)$
• A median from vertex $A(2, 5)$ intersects the midpoint of the opposite side at $M(6.5, 6.5)$.

What is a median in a triangle?

A median of a triangle is a line segment joining a vertex to the midpoint of the opposite side.

Approach:

The median from $A(2, 5)$ runs to the midpoint of the side opposite to $A$, which means $M(6.5, 6.5)$ is the midpoint of the line joining vertices $B(5, 9)$ and the unknown third vertex, say $C(x, y)$.

Step 1: Use the midpoint formula.

The midpoint $M$ of a line segment joining two points $(x_1, y_1)$ and $(x_2, y_2)$ is given by: $M = \left( \frac{x_1 + x_2}{2}, \frac{y_1 + y_2}{2} \right)$

Here, $M(6.5, 6.5)$ is the midpoint of $B(5, 9)$ and the unknown vertex $C(x, y)$.

Thus, we can set up the following equations for the midpoint: $\frac{5 + x}{2} = 6.5 \quad \text{(for the x-coordinate)}$ $\frac{9 + y}{2} = 6.5 \quad \text{(for the y-coordinate)}$

Step 2: Solve for $x$ and $y$.

• For the x-coordinate: $\frac{5 + x}{2} = 6.5 \implies 5 + x = 13 \implies x = 8$

• For the y-coordinate: $\frac{9 + y}{2} = 6.5 \implies 9 + y = 13 \implies y = 4$

Conclusion:

The coordinates of the third vertex $C$ are $(8, 4)$.

Thus, the correct answer is: $\boxed{(8, 4)}$

Would you like more details or have any other questions?

Related Questions:

1. How do you find the centroid of a triangle in a coordinate plane?
2. What are the properties of medians in any triangle?
3. How can you calculate the length of a median in a triangle?
4. How do you derive the midpoint formula in coordinate geometry?
5. How can you prove that the medians of a triangle intersect at a single point?

Tip:

The point where all the medians of a triangle intersect is called the centroid, and it divides each median into a 2:1 ratio.