Given the coordinates of a triangle, A (-3,-4), B (5,-2), C (-5,6)
1. Find the midpoints of each side of the triangle
2. Find the distances of each of the medians of the triangle (the line segment joining the vertex of a triangle and the midpoint of the opposite side)
The midpoint of each line segment is the average of its endpoints.
M(AB) = (A+B)/2 = ((-3+5)/2, (-4-2)/2) = (1,-3)
Do the others in the same way.
Then use your distance formula to find the length of each median.
The length of the median from C to M(AB) is
√((1+5)^2 + (-3-6)^2) = √(36+81) = √117
!. A(-3, -4), M(x, y), B(5, -2).
x + 3 = (5+3)/2,
X = 1.
y+4 = (-2+4)/2,
Y = -3.
B(5, -2), M(x, y), C(-5, 6).
x-5 = (-5-5)/2,
X = 0.
y+2 = (6+ 2)/2,
Y = 2.
A(-3, -4), M(x, y), C(-5, 6).
2. M(1, -3), C(-5, 6).
(MC)^2 = (-5-1)^2 + (6+3) = 36 + 81 = 117,
MC = 10.8.
M(0, 2), A(-3, -4),
(MA)^2 = (-3-0)^2 + (-4-2)^2 = 9 + 36 = 45.
MA = 6.7.
M(x, y), B(5, -2).
To find the midpoints of each side of the triangle, you can use the formula:
Midpoint = ((x1 + x2) / 2, (y1 + y2) / 2)
Let's calculate the midpoints for each side of the triangle:
1. Midpoint of AB:
A(-3, -4), B(5, -2)
Midpoint_AB = ((-3 + 5) / 2, (-4 + (-2)) / 2)
= (2 / 2, -6 / 2)
= (1, -3)
2. Midpoint of BC:
B(5, -2), C(-5, 6)
Midpoint_BC = ((5 + (-5)) / 2, (-2 + 6) / 2)
= (0 / 2, 4 / 2)
= (0, 2)
3. Midpoint of AC:
A(-3, -4), C(-5, 6)
Midpoint_AC = ((-3 + (-5)) / 2, (-4 + 6) / 2)
= (-8 / 2, 2 / 2)
= (-4, 1)
Now we can find the distances of each of the medians of the triangle using the distance formula:
Distance = √((x2 - x1)^2 + (y2 - y1)^2)
Let's calculate the distances for each of the medians:
1. Median from A:
A(-3, -4), Midpoint_BC(0, 2)
Distance_A = √((0 - (-3))^2 + (2 - (-4))^2)
= √((3)^2 + (6)^2)
= √(9 + 36)
= √(45)
2. Median from B:
B(5, -2), Midpoint_AC(-4, 1)
Distance_B = √((-4 - 5)^2 + (1 - (-2))^2)
= √((-9)^2 + (3)^2)
= √(81 + 9)
= √(90)
3. Median from C:
C(-5, 6), Midpoint_AB(1, -3)
Distance_C = √((1 - (-5))^2 + (-3 - 6)^2)
= √((6)^2 + (-9)^2)
= √(36 + 81)
= √(117)
Therefore, the distances of each of the medians of the triangle are √45, √90, and √117, respectively.