show that the sum of the squares of the lengths of the medians of triangle equals three-fourths the sum of the squares of the lengths of the sides.(hint:place the triangle so that its vertices are at points(-a,0),(b0)and (0,c))
To prove that the sum of the squares of the lengths of the medians of a triangle is equal to three-fourths the sum of the squares of the lengths of the sides, we can follow these steps:
Step 1: Set up the triangle
Place the triangle on a coordinate system so that its vertices are at points (-a, 0), (b, 0), and (0, c). This will help us calculate the lengths of the medians and the sides of the triangle.
Step 2: Calculate the lengths of the medians
The medians of a triangle are the line segments connecting each vertex to the midpoint of the opposite side. In this case, we will calculate the lengths of the medians ma, mb, and mc.
Step 3: Calculate the lengths of the sides
Using the coordinates of the vertices, we can calculate the lengths of the sides of the triangle. The sides are given by the distance formula:
ab = √[(b - (-a))^2 + (0 - 0)^2]
ac = √[(0 - (-a))^2 + (c - 0)^2]
bc = √[(b - 0)^2 + (c - 0)^2]
Step 4: Square each length
Square each of the lengths of the medians and sides obtained in the previous steps.
Step 5: Sum the squares of the medians and sides
Add up the squares of the lengths of the medians (ma^2 + mb^2 + mc^2) and the squares of the lengths of the sides (ab^2 + ac^2 + bc^2).
Step 6: Compare the sums
If the sum of the squares of the medians (ma^2 + mb^2 + mc^2) is equal to three-fourths of the sum of the squares of the sides (3/4 * (ab^2 + ac^2 + bc^2)), then the statement is proven.
By following these steps, you should be able to demonstrate that the sum of the squares of the lengths of the medians of a triangle is equal to three-fourths the sum of the squares of the lengths of the sides.