To find the coordinates of point \( P \) on the median that starts at vertex \( (b, c) \) and ends at midpoint \( \left( a_2, 0 \right) \), we will first determine the coordinates of the midpoint between the two endpoints.
The median is the line segment that connects a vertex of a triangle to the midpoint of the opposite side. The median from vertex \( (b, c) \) to the midpoint \( \left( a_2, 0 \right) \) can be expressed parametrically using a parameter \( t \), where \( t \) is typically in the range \( [0, 1] \):
\[ P(t) = (1-t)(b, c) + t(a_2, 0) \]
Breaking this down into its components, we get:
\[ P(t) = \left((1-t)b + ta_2, (1-t)c + t \cdot 0\right) \]
This simplifies to:
\[ P(t) = \left((1-t)b + ta_2, (1-t)c\right) \]
The coordinates of point \( P \) depend on the value of parameter \( t \):
- When \( t = 0 \), \( P(0) = (b, c) \) (the vertex).
- When \( t = 1 \), \( P(1) = (a_2, 0) \) (the midpoint of the opposite side).
If we're looking for the specific coordinates of \( P \) at a particular point along the median, we can substitute the desired value of \( t \) between 0 and 1 into this equation to find \( P(t) \).
For example, if you want to find the midpoint \( P \) on the median, you would use \( t = 0.5 \):
\[ P(0.5) = \left(0.5b + 0.5a_2, 0.5c\right) \]
In summary, the coordinates of \( P \) on the median from \( (b, c) \) to \( \left(a_2, 0\right) \) are given by:
\[ P(t) = \left((1-t)b + ta_2, (1-t)c\right) \]
You can decide the value of \( t \) based on where you need \( P \) to fall along the median.
1 answer