find the lengths of the medians of the triangle with vertices at A(2,-2), B(-4,-4)and C(0,4)
find the midpoint of AB which is (-1,-3)
now find distance between C(0,4) and (-1,-3)
= √((0+1)^2 + (4+3)^2) = √50
do the other two the same way
To find the lengths of the medians of the triangle with vertices A(2,-2), B(-4,-4), and C(0,4), we need to calculate the lengths of the lines connecting each vertex to the midpoint of the opposite side. Here's how you can do it step by step:
Step 1: Find the coordinates of the midpoints of each side.
To find the midpoint of a line segment with endpoints (x₁,y₁) and (x₂,y₂), you can use the midpoint formula:
Midpoint = ((x₁ + x₂) / 2, (y₁ + y₂) / 2)
Midpoint of AB:
x-coordinate = (2 + (-4)) / 2 = -1
y-coordinate = (-2 + (-4)) / 2 = -3
So, the midpoint of AB is (-1,-3).
Midpoint of BC:
x-coordinate = (-4 + 0) / 2 = -2
y-coordinate = (-4 + 4) / 2 = 0
So, the midpoint of BC is (-2,0).
Midpoint of AC:
x-coordinate = (2 + 0) / 2 = 1
y-coordinate = (-2 + 4) / 2 = 1
So, the midpoint of AC is (1,1).
Step 2: Calculate the lengths of the medians.
To find the length of a line segment with endpoints (x₁,y₁) and (x₂,y₂), you can use the distance formula:
Distance = √[(x₂ - x₁)² + (y₂ - y₁)²]
Length of the median from vertex A to the midpoint of BC:
Distance = √[((-2) - 1)² + (0 - (-3))²]
= √[(-3)² + 3²]
= √[9 + 9]
= √18
= 3√2
Length of the median from vertex B to the midpoint of AC:
Distance = √[(1 - (-1))² + (1 - (-3))²]
= √[2² + 4²]
= √[4 + 16]
= √20
= 2√5
Length of the median from vertex C to the midpoint of AB:
Distance = √[((-1) - 2)² + ((-3) - (-2))²]
= √[(-3)² + 1²]
= √[9 + 1]
= √10
So, the lengths of the medians of the triangle are:
- From vertex A to the midpoint of BC: 3√2
- From vertex B to the midpoint of AC: 2√5
- From vertex C to the midpoint of AB: √10