Answer:
To find the lengths of the medians of the triangle, we can use the midpoint formula to find the midpoint of each side, and then calculate the distance from each midpoint to the opposite vertex. Let's start by finding the midpoints of the sides:
1. Midpoint of side 1 (between vertices (1, -1) and (0, 4)):
Midpoint = ((1 + 0) / 2, (-1 + 4) / 2) = (0.5, 1.5)
2. Midpoint of side 2 (between vertices (0, 4) and (-5, 3)):
Midpoint = ((0 - 5) / 2, (4 + 3) / 2) = (-2.5, 3.5)
3. Midpoint of side 3 (between vertices (-5, 3) and (1, -1)):
Midpoint = ((-5 + 1) / 2, (3 - 1) / 2) = (-2, 1)
Now, we calculate the lengths of the medians, which are the distances from each midpoint to the opposite vertex:
1. Median 1: From midpoint (0.5, 1.5) to vertex (-5, 3)
Length = √[(0.5 - (-5))^2 + (1.5 - 3)^2]
2. Median 2: From midpoint (-2.5, 3.5) to vertex (1, -1)
Length = √[(-2.5 - 1)^2 + (3.5 - (-1))^2]
3. Median 3: From midpoint (-2, 1) to vertex (0, 4)
Length = √[(-2 - 0)^2 + (1 - 4)^2]
You can calculate these distances to find the lengths of the medians. Let me know if you need further assistance with the calculations.
Answer:
Absolutely, we can find the lengths of the medians in the triangle with vertices (1, -1), (0, 4), and (-5, 3). Here's how:
1. Find the coordinates of the medians' endpoints:
A median of a triangle connects a vertex to the midpoint of the opposite side. Let's find the midpoints of each side:
Midpoint of side BC (between (0, 4) and (-5, 3)):
x-coordinate: ((0 - 5) / 2) = -2.5
y-coordinate: ((4 + 3) / 2) = 3.5
Midpoint of BC = (-2.5, 3.5)
Midpoint of side AC (between (1, -1) and (-5, 3)):
x-coordinate: ((1 - 5) / 2) = -2
y-coordinate: ((-1 + 3) / 2) = 1
Midpoint of AC = (-2, 1)
Midpoint of side AB (between (1, -1) and (0, 4)):
x-coordinate: ((1 - 0) / 2) = 0.5
y-coordinate: ((-1 + 4) / 2) = 1.5
Midpoint of AB = (0.5, 1.5)
2. Calculate the lengths of the medians:
Now that we have the endpoints, we can find the lengths of the medians using the distance formula:
Distance formula: d = √((x2 - x1)^2 + (y2 - y1)^2)
Length of median from A (1, -1) to (-2.5, 3.5):
d = √(((-2.5) - (1))^2 + ((3.5) - (-1))^2)
d = √((3.5)^2 + (4.5)^2)
d ≈ √30.25 (approximately 5.70)
Length of median from B (0, 4) to (-2, 1):
d = √(((-2) - (0))^2 + ((1) - (4))^2)
d = √((2)^2 + (-3)^2)
d = √13
Length of median from C (-5, 3) to (0.5, 1.5):
d = √(((0.5) - (-5))^2 + ((1.5) - (3))^2)
d = √((5.5)^2 + (-1.5)^2)
d ≈ √30.25 (approximately 5.70)
Therefore:
Length of median from A: ≈ 5.70
Length of median from B: √13
Length of median from C: ≈ 5.70
