To find the medians of a triangle and verify their intersection point, we first need to identify the midpoints of each side of the triangle. The medians are then the segments that connect each vertex of the triangle to the midpoint of the opposite side.
Given the vertices of a triangle are \(a = (6, 2)\), \(b = (4, 5)\), and \(c = (1, 2)\):
-
Find the midpoints of each side:
-
Midpoint of side \(b\) and \(c\): \[ M_{bc} = \left( \frac{4 + 1}{2}, \frac{5 + 2}{2} \right) = \left( \frac{5}{2}, \frac{7}{2} \right) = (2.5, 3.5) \]
-
Midpoint of side \(a\) and \(c\): \[ M_{ac} = \left( \frac{6 + 1}{2}, \frac{2 + 2}{2} \right) = \left( \frac{7}{2}, 2 \right) = (3.5, 2) \]
-
Midpoint of side \(a\) and \(b\): \[ M_{ab} = \left( \frac{6 + 4}{2}, \frac{2 + 5}{2} \right) = \left( 5, \frac{7}{2} \right) = (5, 3.5) \]
-
-
Identify the ordered pairs: From our calculations of the midpoints, we have:
- Midpoint of side \(b\) (4, 5) and \(c\) (1, 2): \(M_{bc} = (2.5, 3.5)\)
- Midpoint of side \(a\) (6, 2) and \(c\) (1, 2): \(M_{ac} = (3.5, 2)\)
- Midpoint of side \(a\) (6, 2) and \(b\) (4, 5): \(M_{ab} = (5, 3.5)\)
-
Finding the answer: The ordered pairs we found align with response (which need to be corrected according to the calculations):
- \(c = (2.5, 3.5)\)
- \(b = (5, 3.5)\)
- \(a = (3.5, 2)\)
Now, check for matches with the given options. Based on this logic, we see that the set is not an exact match with the responses as provided. However, review them closely to find which of the specified choices can possibly have matching coordinates relevant to the triangle systems you've provided. The choice closest to providing midpoints to answer options resembles:
- Option 5 (c = (2.5,3.5), b = (5,3.5), and a = (3.5, 2)) aligns with the midpoints we calculated.
Thus, the final answer to the original question is closest to:
- c=(2.5,3.5), b=(5,3.5), and a=(3.5,2).
If options differ or are written incorrectly, always double-check the conveyance for total clarity.