The line segment joining a vertex of a triangle and the midpoint of the opposite side is called the median of the triangle. Given a triangle whose vertices are A(4,-4), B(10, 4) and C(2, 6), find the point on each median that is two-thirds of the distance from the vertex to the midpoint of the opposite side.
Since all three medians meet there, finding where two intersect is enough.
AB has midpoint P at (5,0)
BC has midpoint Q at (6,5)
The lines CP and AQ are
y-6 = (0-6)/(5-2) (x-2)
y+4 = (5+4)/(6-4) (x-4)
Now just find where the two lines intersect.
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To find the point on each median that is two-thirds of the distance from the vertex to the midpoint of the opposite side, we need to follow these steps:
1. Find the midpoint of each side.
2. Calculate the distance from each vertex to the midpoint of the opposite side.
3. Multiply the distance by 2/3.
4. Find the point on the median by adding the vector formed by the midpoint to the vertex, scaled by the distance calculated in step 3.
Let's go through each step to find the points:
Step 1: Find the midpoint of each side.
- The midpoint of AB can be found by averaging the x-coordinates and the y-coordinates separately:
- Midpoint_AB = ((x1 + x2)/2, (y1 + y2)/2) = ((4 + 10)/2, (-4 + 4)/2) = (7, 0)
- The midpoint of AC:
- Midpoint_AC = ((x1 + x3)/2, (y1 + y3)/2) = ((4 + 2)/2, (-4 + 6)/2) = (3, 1)
- The midpoint of BC:
- Midpoint_BC = ((x2 + x3)/2, (y2 + y3)/2) = ((10 + 2)/2, (4 + 6)/2) = (6, 5)
Step 2: Calculate the distance from each vertex to the midpoint of the opposite side.
- The distance from A to Midpoint_BC can be found using the distance formula:
- Distance_AC = √((x2 - x1)^2 + (y2 - y1)^2)
- Distance_AC = √((6 - 4)^2 + (5 - (-4))^2) = √(2^2 + 9^2) = √(4 + 81) = √85
- Similarly, the distance from B to Midpoint_AC is also √85, and the distance from C to Midpoint_AB is √72.
Step 3: Multiply the distance by 2/3.
- For point A, the distance to Midpoint_BC is √85, so 2/3 of that is (2/3) * √85.
- For point B, the distance to Midpoint_AC is √85, so 2/3 of that is (2/3) * √85.
- For point C, the distance to Midpoint_AB is √72, so 2/3 of that is (2/3) * √72.
Step 4: Find the point on the median by adding the vector formed by the midpoint to the vertex, scaled by the distance calculated in step 3.
- For point A, we add (7, 0) + (4, -4) * (2/3) * √85.
- For point B, we add (6, 5) + (10, 4) * (2/3) * √85.
- For point C, we add (3, 1) + (2, 6) * (2/3) * √72.
By performing these calculations, we can find the three points on each median that are two-thirds of the distance from the vertex to the midpoint of the opposite side in the given triangle.