LMN have vertices at L(0,4) M(-5,2),and N (2,-2). determine the equation for the perpendicular bisector that passes through MN
M(-5,2), (x,y), N(2,-2).
X = (-5+2) / 2 = -3/2.
Y = (2+(-2)) / 2 = 0.
m1 = (-2-2) / (2-(-5)) = -4/7.
m2 = 7/4 = Negative reciprocal of m1.
(-3/2,0). m2 = 7/4.
Y = mx + b.
(7/4)(-3/2) + b = 0.
b = 21/8.
Eq: Y = (7/4)x + 21/8.
Oh, finding equations for lines? Let's get this party started! 🎉
To find the equation for the perpendicular bisector, we first need to calculate the midpoint of line segment MN. The midpoint formula is just a fancy way of saying, "Hey, let's average the x-coordinates and the y-coordinates!"
Midpoint x-coordinate (x_m) = (x_1 + x_2) / 2
Midpoint y-coordinate (y_m) = (y_1 + y_2) / 2
Using the coordinates of M(-5, 2) and N(2, -2), we can calculate the midpoint:
x_m = (-5 + 2) / 2 = -3/2
y_m = (2 + (-2)) / 2 = 0
So the midpoint of MN is (-3/2, 0). Looking spiffy so far! 😎
Next, let's find the slope of line MN using the formula:
slope (m) = (y_2 - y_1) / (x_2 - x_1)
Using the coordinates of M(-5, 2) and N(2, -2), we get:
m = (-2 - 2) / (2 - (-5)) = -4/7
Hold onto your funny bone, the punchline is coming! 😄
Since the perpendicular bisector has the opposite reciprocal slope, the slope of the perpendicular bisector (m_perpendicular) is:
m_perpendicular = -1 / m
Now let's substitute in the value for m:
m_perpendicular = -1 / (-4/7) = 7/4
So far so good! Let's keep the laughter rolling! 😂
To find the equation for the perpendicular bisector, we'll use the point-slope form of a line:
(y - y_m) = m_perpendicular(x - x_m)
Plugging in the values for the midpoint (-3/2, 0) and the perpendicular slope (7/4), we get:
(y - 0) = (7/4)(x - (-3/2))
y = (7/4)(x + 3/2)
And there you have it, folks! The equation for the perpendicular bisector that passes through MN is y = (7/4)(x + 3/2). Enjoy the show! 🤡🎭
To find the equation for the perpendicular bisector of MN, we need to follow these steps:
Step 1: Find the midpoint of MN.
Step 2: Find the slope of MN.
Step 3: Find the negative reciprocal of the slope from Step 2 to get the slope of the perpendicular bisector.
Step 4: Use the midpoint from Step 1 and the slope from Step 3 in the point-slope form equation to find the equation of the perpendicular bisector.
Let's go through each step:
Step 1: Find the midpoint of MN.
The midpoint formula is:
Midpoint = ((x1 + x2) / 2, (y1 + y2) / 2)
Given the coordinates of M(-5,2) and N(2,-2), we can calculate the midpoint:
Midpoint = ((-5 + 2) / 2, (2 + (-2)) / 2)
Midpoint = (-3 / 2, 0)
So the midpoint of MN is (-3/2, 0).
Step 2: Find the slope of MN.
The slope formula is:
Slope = (y2 - y1) / (x2 - x1)
Given the coordinates of M(-5,2) and N(2,-2), we can calculate the slope:
Slope = (-2 - 2) / (2 - (-5))
Slope = -4 / 7
So the slope of MN is -4/7.
Step 3: Find the negative reciprocal of the slope from Step 2 to get the slope of the perpendicular bisector.
The negative reciprocal of -4/7 is 7/4.
So the slope of the perpendicular bisector is 7/4.
Step 4: Use the midpoint from Step 1 and the slope from Step 3 in the point-slope form equation to find the equation of the perpendicular bisector.
The point-slope form equation is:
y - y1 = m(x - x1)
Using the values from Step 1 and Step 3, the equation of the perpendicular bisector is:
y - 0 = (7/4)(x - (-3/2))
Simplifying this equation:
y = (7/4)(x + 3/2)
Thus, the equation for the perpendicular bisector that passes through MN is:
y = (7/4)x + 21/8
To find the equation for the perpendicular bisector that passes through line segment MN, you need to find the midpoint of MN first and then determine the slope of the perpendicular bisector.
1. Find the midpoint of MN:
The midpoint of a line segment is calculated by finding the averages of the x-coordinates and y-coordinates of the endpoints.
Midpoint formula:
X = (x1 + x2) / 2
Y = (y1 + y2) / 2
In this case, the coordinates of M are (-5,2) and the coordinates of N are (2,-2).
Therefore, the midpoint coordinates (x, y) are:
X = (-5 + 2) / 2 = -3/2
Y = (2 + (-2)) / 2 = 0
So, the midpoint of MN is (-3/2, 0).
2. Find the slope of MN:
Slope of a line is calculated using the formula:
m = (y2 - y1) / (x2 - x1)
In this case, the coordinates of M are (-5,2) and the coordinates of N are (2,-2).
So, the slope of MN is:
m = (y2 - y1) / (x2 - x1) = (-2 - 2) / (2 - (-5)) = -4/7
3. Calculate the negative reciprocal of the slope:
The perpendicular bisector of a line has a slope that is the negative reciprocal of the original line's slope.
The negative reciprocal of -4/7 is 7/4.
4. Find the equation of the perpendicular bisector using the point-slope form:
The point-slope form of a line is given by:
y - y1 = m(x - x1)
Using the midpoint (-3/2, 0) and the slope 7/4:
y - 0 = (7/4)(x - (-3/2))
y = (7/4)(x + 3/2)
Simplifying:
y = (7/4)x + (21/8)
So, the equation for the perpendicular bisector that passes through MN is y = (7/4)x + (21/8).