Answer:
(a) The equation of AD is y = 4.5·x -7.5
The equation of BE is y = -0.3·x + 0.5
The equation of CF is y = -1.5·x + 2.5.
(b) The three equations representing the three lines AD, BE and CF, pass through the same point (which can be written as (1.667, 0)).
Step-by-step explanation:
The given triangle coordinates are;
A = (3, 6)
B = (-5, 2)
C = (7, -8)
The midpoint of BC = D
The midpoint of AC = E
The midpoint of AB = F
Therefore, we have;
The coordinates of point D = (B + C)/2 = (((-5) + 7 )/2, (2 + (-8))/2)
The coordinates of point D = (1, -3)
The coordinates of point E = (A + C)/2 = ((3 + 7 )/2, (6 + (-8))/2)
The coordinates of point E = (5, -1)
The coordinates of point F = (A + B)/2 = ((3 + (-5) )/2, (6 + 2)/2)
The coordinates of point F = (-1, 4)
The equation of AD in slope and intercept form is therefore;
A = (3, 6), D = (1, -3)
Therefore, we have;
y - 6 = 4.5×(x - 3)
y - 6 = 4.5·x -13.5
y = 4.5·x -13.5 + 6 = 4.5·x -7.5
y = 4.5·x -7.5
The equation of BE in slope and intercept form is found as follows;
B = (-5, 2), E = (5, -1)
Slope = ((-1)-2)/(5 - (-5)) = -0.3
The point slope equation is y - 2 = -0.3×(x - (-5))
y - 2 = -0.3·x - 1.5
y = -0.3·x - 1.5 + 2 = -0.3·x + 0.5
y = -0.3·x + 0.5
The equation of CF in slope and intercept form is found as follows;
C = (7, -8), F = (-1, 4)
Slope = (4 - (-8))/((-1) - 7) = -1.5
The point slope equation is y - (-8) = -1.5×(x - 7)
y + 8 = -1.5·x + 10.5
y = -1.5·x + 10.5 - 8
y = -1.5·x + 2.5.
Where CF and BE intersect, we have;
-0.3·x + 0.5 = -1.5·x + 2.5.
1.5·x -0.3·x = 2.5 - 0.5 = 2.0
x = 2.0/1.2 = 5/3
y = -1.5×5/3+ 2.5 = 0
We check for line AD, where y = 4.5·x -7.5
When x = 5/3, we get;
y = 4.5×(5/3) -7.5 = 0
Therefore, the three equations pass through the point (5/3, 0) which is the same point.