given triangle ABC with A(-3,4) B (5,8) and C(2,-2). write the equation of the line containing midsegment Line XZ in standard form, where X is the midpoint of Line AB and Z is the midpoint of Line BC
To find the equation of the line containing the midsegment Line XZ, we need to find the coordinates of points X and Z.
First, we find the coordinates of point X, which is the midpoint of line AB. The coordinates of A are (-3, 4) and the coordinates of B are (5, 8). The midpoint formula is given by:
X = ((x1 + x2)/2, (y1 + y2)/2)
For point X, the x-coordinate is:
x_X = (-3 + 5)/2 = 2/2 = 1
And the y-coordinate is:
y_X = (4 + 8)/2 = 12/2 = 6
Therefore, the coordinates of point X are X(1, 6).
Next, we find the coordinates of point Z, which is the midpoint of line BC. The coordinates of B are (5, 8) and the coordinates of C are (2, -2). Using the midpoint formula again:
Z = ((x2 + x3)/2, (y2 + y3)/2)
For point Z, the x-coordinate is:
x_Z = (5 + 2)/2 = 7/2 = 3.5
And the y-coordinate is:
y_Z = (8 - 2)/2 = 6/2 = 3
Therefore, the coordinates of point Z are Z(3.5, 3).
Now that we have the coordinates of points X and Z, we can write the equation of the line containing XZ in standard form, which is Ax + By = C.
To find the coefficients A, B, and C, we can use the point-slope form of a linear equation:
y - y_1 = m(x - x_1)
where m is the slope of line XZ and (x_1, y_1) is a point on the line. Here, we can use point X(1, 6) or point Z(3.5, 3) to find the slope.
Using point X(1, 6):
slope (m) = (y - y_X)/(x - x_X) = (y - 6)/(x - 1)
Substituting the coordinates of point Z(3.5, 3):
m = (3 - 6)/(3.5 - 1) = -3/2.5 = -6/5
Now, using the slope-intercept form, which is y = mx + b, with the slope m = -6/5, we can calculate the y-intercept b:
6 = (-6/5)(1) + b
6 = -6/5 + b
b = 6 + 6/5 = 36/5 + 6/5 = 42/5
So, the equation of the line containing the midsegment Line XZ in slope-intercept form is:
y = (-6/5)x + 42/5
To convert the equation to standard form, we multiply every term by 5 to eliminate the denominator:
5y = -6x + 42
Rearranging the terms:
6x + 5y = 42
Therefore, the equation of the line containing the midsegment Line XZ in standard form is 6x + 5y = 42.