Finding the Centroid of a Triangle Given a Vertex and Two Midpoints

This problem involves coordinate geometry and understanding the properties of a triangle, specifically the concepts of midpoints and the centroid. We are given one vertex of the triangle and the midpoints of the two sides that meet at this vertex. Our goal is to find the coordinates of the triangle's centroid.

Understanding the Given Information

• Let the triangle be denoted as \(\triangle ABC\).
• One vertex is given as \(A = (1, 1)\).
• The midpoints of the two sides through vertex \(A\) are given as \(D = (-1, 2)\) and \(E = (3, 2)\). Let's assume \(D\) is the midpoint of side \(AB\) and \(E\) is the midpoint of side \(AC\).

Step-by-Step Solution

Step 1: Find the Coordinates of Vertex B

Since \(D(-1, 2)\) is the midpoint of the side \(AB\), we can use the midpoint formula. Let the coordinates of \(B\) be \((x_B, y_B)\). The midpoint formula for a segment with endpoints \((x_1, y_1)\) and \((x_2, y_2)\) is \(\left(\frac{x_1+x_2}{2}, \frac{y_1+y_2}{2}\right)\).

Using the midpoint formula for segment \(AB\) with \(A(1, 1)\) and \(D(-1, 2)\):

• \(x\)-coordinate: \(-1 = \frac{1 + x_B}{2}\)
• \(y\)-coordinate: \(2 = \frac{1 + y_B}{2}\)

Solving for \(x_B\) and \(y_B\):

• \(-1 \times 2 = 1 + x_B \Rightarrow -2 = 1 + x_B \Rightarrow x_B = -2 - 1 = -3\).
• \(2 \times 2 = 1 + y_B \Rightarrow 4 = 1 + y_B \Rightarrow y_B = 4 - 1 = 3\).

So, the coordinates of vertex \(B\) are \((-3, 3)\).

Step 2: Find the Coordinates of Vertex C

Similarly, since \(E(3, 2)\) is the midpoint of the side \(AC\), we use the midpoint formula. Let the coordinates of \(C\) be \((x_C, y_C)\).

Using the midpoint formula for segment \(AC\) with \(A(1, 1)\) and \(E(3, 2)\):

• \(x\)-coordinate: \(3 = \frac{1 + x_C}{2}\)
• \(y\)-coordinate: \(2 = \frac{1 + y_C}{2}\)

Solving for \(x_C\) and \(y_C\):

• \(3 \times 2 = 1 + x_C \Rightarrow 6 = 1 + x_C \Rightarrow x_C = 6 - 1 = 5\).
• \(2 \times 2 = 1 + y_C \Rightarrow 4 = 1 + y_C \Rightarrow y_C = 4 - 1 = 3\).

So, the coordinates of vertex \(C\) are \((5, 3)\).

Step 3: Find the Coordinates of the Centroid

Now that we have the coordinates of all three vertices of the triangle: \(A(1, 1)\), \(B(-3, 3)\), and \(C(5, 3)\), we can find the centroid. The centroid of a triangle with vertices \((x_1, y_1)\), \((x_2, y_2)\), and \((x_3, y_3)\) is given by the formula \(\left(\frac{x_1+x_2+x_3}{3}, \frac{y_1+y_2+y_3}{3}\right)\).

Using the centroid formula for \(\triangle ABC\):

• \(x\)-coordinate of centroid: \(\frac{1 + (-3) + 5}{3} = \frac{1 - 3 + 5}{3} = \frac{3}{3} = 1\).
• \(y\)-coordinate of centroid: \(\frac{1 + 3 + 3}{3} = \frac{7}{3}\).

So, the centroid of the triangle is \(\left(1, \frac{7}{3}\right)\).

Summary of Coordinates

The calculated centroid coordinates are \(\left(1, \frac{7}{3}\right)\).

Conclusion

By using the midpoint formula to find the other two vertices from the given vertex and the midpoints of the adjacent sides, we were able to determine the coordinates of all three vertices. Then, applying the centroid formula to these three vertices yielded the coordinates of the centroid of the triangle.

Revision Table: Triangle Geometry Formulas

Additional Information: Properties of Triangle Centroid

• The centroid divides each median in a 2:1 ratio, with the longer segment being between the vertex and the centroid.
• The centroid is always inside the triangle.
• A median is a line segment joining a vertex to the midpoint of the opposite side.
• The three medians of a triangle intersect at a single point, which is the centroid.