A triangle shows points A(2,8), B(14,4) and C(4,2). M is the midpoint of AC.
a. Write down the coordinates of M
b. Calculate the gradient of line AB.
c. Find the equation of the line parallel to AB that passes through M.
geez, impatient much?
a. M= [x1+x2/2,y1+y2/2] A=[2,8] C= [4,2]
=[2+4/2,8+2/2] = [6/3,10/2]= [3,5]
b. slope = y2-y1/x2-x1 A =[2,8] B = 14,4]
= 4-8/14-2 = -4/12 = -1/3
c. slope = -1/3, passing through [3,5]
equation of the line = [y-5] = -1/3[x-3]
= 3[y-5] = -1[x-3]
= 3y-15 = -x+3
= x+ 3y -18= 0
To solve this problem, we need to go step by step.
a. The midpoint formula states that the coordinates of the midpoint M between two points A(x1, y1) and C(x2, y2) are given by:
M = ((x1 + x2) / 2, (y1 + y2) / 2)
Using the given coordinates A(2, 8) and C(4, 2), we substitute these values into the midpoint formula:
M = ((2 + 4) / 2, (8 + 2) / 2)
M = (6 / 2, 10 / 2)
M = (3, 5)
Therefore, the coordinates of the midpoint M are (3, 5).
b. The gradient of a line AB can be calculated using the gradient formula:
Gradient (m) = (y2 - y1) / (x2 - x1)
Using the coordinates A(2, 8) and B(14, 4), we substitute these values into the gradient formula:
Gradient (m) = (4 - 8) / (14 - 2)
Gradient (m) = -4 / 12
Gradient (m) = -1/3
Therefore, the gradient of line AB is -1/3.
c. To find the equation of the line parallel to AB that passes through M, we can use the point-slope form of a linear equation.
The equation of a line parallel to AB can be written as:
y - y1 = m(x - x1)
Where (x1, y1) represents the coordinates of point M and m represents the gradient of line AB.
Using the coordinates of M(3, 5) and the gradient of AB (-1/3), we substitute these values into the point-slope form:
y - 5 = (-1/3)(x - 3)
Rearranging the equation, we get:
3(y - 5) = -1(x - 3)
Expanding, we get:
3y - 15 = -x + 3
Simplifying, we get:
x + 3y = 18
Therefore, the equation of the line parallel to AB that passes through M is x + 3y = 18.