Key Concepts and Formulas

• Median and Centroid: A median connects a vertex to the midpoint of the opposite side. The centroid is the intersection point of the medians, dividing each median in a 2:1 ratio. The centroid divides the triangle into three smaller triangles of equal area.
• Midpoint Formula: The midpoint of a line segment with endpoints $(x_1, y_1)$ and $(x_2, y_2)$ is given by $\left(\frac{x_1 + x_2}{2}, \frac{y_1 + y_2}{2}\right)$.
• Area of a Triangle (Coordinate Formula): The area of a triangle with vertices $(x_1, y_1)$, $(x_2, y_2)$, and $(x_3, y_3)$ is given by $\frac{1}{2} |x_1(y_2 - y_3) + x_2(y_3 - y_1) + x_3(y_1 - y_2)|$.

Step-by-Step Solution

Step 1: Understand the given information and deduce C's x-coordinate.

We are given vertex A(1, 2), the median from B is $x + y = 5$, and the median from C is $x = 4$. Since the median from C has the equation $x = 4$, and this median passes through vertex C, the x-coordinate of C must be 4. Let C = (4, $y_c$).

Why this step? We want to find the coordinates of all vertices. The given information directly gives us the x-coordinate of vertex C.

Step 2: Find the y-coordinate of vertex C.

Let D be the midpoint of AC. Then D lies on the median from B, i.e., the line $x + y = 5$.

• Find the midpoint D of AC: $D = \left(\frac{1+4}{2}, \frac{2+y_c}{2}\right) = \left(\frac{5}{2}, \frac{2+y_c}{2}\right)$.
• Since D lies on $x + y = 5$, we have $\frac{5}{2} + \frac{2+y_c}{2} = 5$.
• Multiplying by 2, we get $5 + 2 + y_c = 10$, so $7 + y_c = 10$, which implies $y_c = 3$.

Therefore, C = (4, 3).

Why this step? We use the fact that the midpoint of AC lies on the median from B to find the y-coordinate of C.

Step 3: Find the coordinates of the Centroid G.

The centroid G is the intersection of the medians. We have the equations of two medians: $x + y = 5$ and $x = 4$. Substituting $x = 4$ into the first equation, we get $4 + y = 5$, so $y = 1$. Thus, G = (4, 1).

Why this step? The centroid is crucial for relating the area of $\Delta ABC$ to the area of $\Delta GCA$.

Step 4: Calculate the area of triangle GCA.

We have G(4, 1), C(4, 3), and A(1, 2). Using the coordinate area formula: $Area(\Delta GCA) = \frac{1}{2} |4(3-2) + 4(2-1) + 1(1-3)| = \frac{1}{2} |4(1) + 4(1) + 1(-2)| = \frac{1}{2} |4 + 4 - 2| = \frac{1}{2} |6| = 3.$

Why this step? We calculate the area of $\Delta GCA$ using the coordinates of G, C, and A.

Step 5: Calculate the area of triangle ABC.

Since the centroid divides the triangle into three triangles of equal area, we have $Area(\Delta ABC) = 3 \times Area(\Delta GCA) = 3 \times 3 = 9$.

Why this step? We use the property that the centroid divides the triangle into three equal areas to find the area of the entire triangle.

Common Mistakes & Tips

• Be careful with signs when using the area formula.
• Remember that the centroid divides each median in a 2:1 ratio.
• The centroid divides the triangle into three triangles of equal area.

Summary

We used the given median equations and the properties of the centroid to find the coordinates of vertex C and the centroid G. Then, we calculated the area of $\Delta GCA$ and used the fact that the area of $\Delta ABC$ is three times the area of $\Delta GCA$ to find the area of $\Delta ABC$.

The final answer is $\boxed{9}$, which corresponds to option (D).