given abc with a(-4,1), b(8,-1), and c(-2,9), write the equation of the line containing midsegment xz in standard form, where x is the midpoint of ab and z is the midpoint of bc.
Well, let's find the midpoint of AB first. Using the midpoint formula, the x-coordinate of the midpoint (X) is (-4 + 8)/2 = 2, and the y-coordinate (Y) is (1 + (-1))/2 = 0.
Now let's find the midpoint of BC. Again, using the midpoint formula, the x-coordinate of the midpoint (Z) is (8 + (-2))/2 = 3, and the y-coordinate (W) is (-1 + 9)/2 = 4.
Now, we have the two points X(2, 0) and Z(3, 4), which define the midsegment XZ. To find the equation in standard form, we need to determine the slope (m) and the y-intercept (b) of the line.
The slope (m) can be calculated as (W - Y)/(Z - X) = (4 - 0)/(3 - 2) = 4.
To find the y-intercept (b), we can substitute the coordinates of one of the points (X or Z) into the equation Y = mX + b. I'll use the coordinates of point X, which are (2, 0):
0 = (4)(2) + b
0 = 8 + b
b = -8
So, we have the slope (m = 4) and the y-intercept (b = -8).
Therefore, the equation of the line containing the midsegment XZ is y = 4x - 8 in standard form.
To find the equation of the line containing midsegment XZ, where X is the midpoint of AB and Z is the midpoint of BC, we need to find the coordinates of X and Z first.
Step 1: Find the coordinates of the midpoint X of AB:
The midpoint formula is given by:
Midpoint X = ((x₁ + x₂) / 2, (y₁ + y₂) / 2)
Given the coordinates A(-4,1) and B(8,-1), the coordinates of X are:
Midpoint X = ((-4 + 8) / 2, (1 + (-1)) / 2)
Midpoint X = (4 / 2, 0 / 2)
Midpoint X = (2, 0)
Step 2: Find the coordinates of the midpoint Z of BC:
The midpoint formula is given by:
Midpoint Z = ((x₁ + x₂) / 2, (y₁ + y₂) / 2)
Given the coordinates B(8,-1) and C(-2,9), the coordinates of Z are:
Midpoint Z = ((8 + (-2)) / 2, (-1 + 9) / 2)
Midpoint Z = (6 / 2, 8 / 2)
Midpoint Z = (3, 4)
Step 3: Find the equation of the line containing XZ:
We can find the slope of the line using the coordinates of X and Z and then use the point-slope form of a linear equation to find the equation in standard form.
The slope (m) of the line is given by:
m = (y₂ - y₁) / (x₂ - x₁)
Using the coordinates of X(2,0) and Z(3,4), we have:
m = (4 - 0) / (3 - 2)
m = 4 / 1
m = 4
Now, let's use the point-slope form of a linear equation:
y - y₁ = m(x - x₁)
Using the point X(2,0):
y - 0 = 4(x - 2)
y = 4x - 8
Now, let's write the equation in standard form:
4x - y = 8
Therefore, the equation of the line containing midsegment XZ in standard form is 4x - y = 8.
To find the equation of the line containing the midsegment XZ, we need to find the coordinates of X and Z first.
The midpoint formula states that the coordinates of the midpoint of a line segment with endpoints (x₁, y₁) and (x₂, y₂) are given by:
X = ((x₁ + x₂)/2, (y₁ + y₂)/2)
For our problem, X is the midpoint of AB with endpoints A(-4,1) and B(8,-1). Using the midpoint formula, we can find X:
X = ((-4 + 8)/2, (1 + -1)/2)
X = (4/2, 0/2)
X = (2, 0)
Similarly, Z is the midpoint of BC with endpoints B(8,-1) and C(-2,9):
Z = ((8 + -2)/2, (-1 + 9)/2)
Z = (6/2, 8/2)
Z = (3, 4)
Now that we have the coordinates of X and Z, we can find the equation of the line containing XZ. We'll use the point-slope form of a linear equation:
y - y₁ = m(x - x₁)
where (x₁, y₁) is a point on the line, and m is the slope of the line.
To find the slope of XZ, we use the slope formula:
m = (y₂ - y₁)/(x₂ - x₁)
Using the coordinates of X(2, 0) and Z(3, 4):
m = (4 - 0)/(3 - 2)
m = 4/1
m = 4
Now we can use the point-slope form with the coordinates of X and the slope:
y - 0 = 4(x - 2)
Expanding and rearranging the equation gives us the standard form:
y - 0 = 4x - 8
y = 4x - 8
Therefore, the equation of the line containing the midsegment XZ in standard form is y = 4x - 8.
hint:
1) use your midpoint formula to find X, and then the midpoint formula again to find z
2) Use those two points to find the slope of the slope of the new line
3) Use the slope and a point to find the equation of the new line.