Prove that area of triangle formed by the medians of any triangle is 3/4th the area of given triangle
try it with an equilateral triangle
there is something amiss with the question
should probably be 1/4th
To prove that the area of the triangle formed by the medians of any triangle is 3/4th the area of the given triangle, we can use the concept of coordinate geometry.
1. Consider an arbitrary triangle with coordinates A(x1, y1), B(x2, y2), and C(x3, y3).
2. Find the midpoints of the sides of the triangle. The midpoint of AB is M1, the midpoint of BC is M2, and the midpoint of AC is M3. The coordinates of the midpoints can be calculated using the following formulas:
- M1: ((x1 + x2)/2, (y1 + y2)/2)
- M2: ((x2 + x3)/2, (y2 + y3)/2)
- M3: ((x1 + x3)/2, (y1 + y3)/2)
3. Now, find the equations of the lines passing through the midpoints of the sides of the triangle. These lines are parallel to the sides of the original triangle.
- The equation of the line passing through M1 and parallel to BC is given by: y - (y1 + y2)/2 = (x - (x1 + x2)/2) * (y2 - y1)/(x2 - x1)
- Similarly, the equations of the other two lines passing through M2 and M3 can also be found.
4. Find the point of intersection of these three lines. This point is the centroid of the original triangle. Let's denote it as G(xg, yg).
5. Now, we need to find the coordinates of the centroid, G. The centroid is the average of the coordinates of the vertices of the triangle.
- G: ((x1 + x2 + x3)/3, (y1 + y2 + y3)/3)
6. Using the coordinates of G, we can determine the lengths GM1, GM2, and GM3.
- GM1: √[((x1 + x2)/2 - (x1 + x2 + x3)/3)^2 + ((y1 + y2)/2 - (y1 + y2 + y3)/3)^2]
- GM2 and GM3 can be calculated similarly.
7. Calculate the area of the triangle with vertices G, M1, and M2 using the formula for the area of a triangle. Let's denote it as Agm1m2.
8. Calculate the area of the original triangle ABC using its coordinates with the formula for the area of a triangle. Let's denote it as Aabc.
If the area of the triangle formed by the medians of any triangle is indeed 3/4th the area of the given triangle, then we should have Agm1m2 = (3/4) * Aabc.
By following the above steps and performing the calculations, you can verify that Agm1m2 = (3/4) * Aabc, thereby proving that the area of the triangle formed by the medians of any triangle is 3/4th the area of the given triangle.