1. Determine the equation of the right bisector of the line segment E(2, 6) and F(4, -2). Draw the diagram.
2. Given three points D(2, 5) E(-2, -3) and F(4, -6) determine whether or not the line through D and E is perpendicular to the line through E and F.
3. A triangle is formed from the points L(-3, 6), N(3, 2) and P(1, -8). Find the equation of the following lines:
the median from N
the right bisector of LP
the altitude from N
If you
can find the midpoint of a line segment
know that the slope of the perpendicular is -1/(slope of line segment)
remember the point-slope form of a line
then all of these problems are straighforward. Review those topics, give it a try, show where you get stuck, if you do.
1. E(2,6), M(x,y), F(4.-2).
m1 = (-2-6)/(4-2) = -8/2 = -4
m2 is the negative reciprocal of m1:
m2 = 1/4
x = (2+4)/2 = 3
y = (6+(-2))/2 = 2
Y = mx + b = 2
(1/4)*x + b = 2
3/4 + b = 2
b = 2-3/4 = 5/4
Eq: Y = x/4 + 5/4
2. D(2,6), E(-2,-3), F(4,-6).
m1 = (-3-6)/(-2-2) = -9/-4 = 9/4.
m2 = (-6-(-3))/(4-(-2)) = -3/6 = -1/2.
m2 is NOT the negative reciprocal of m1,
therefore, the lines are NOT perpendicular.
1. To determine the equation of the right bisector of a line segment, we need to find the midpoint of the line segment and the slope perpendicular to the line segment.
First, let's find the midpoint of the line segment using the coordinates of points E(2, 6) and F(4, -2). The midpoint formula is given by:
Midpoint (M) = [(x1 + x2) / 2, (y1 + y2) / 2]
Using the coordinates, we get:
M = [(2 + 4) / 2, (6 + (-2)) / 2]
= [3, 2]
Now, let's find the slope of the line segment. The slope formula between two points (x1, y1) and (x2, y2) is given by:
Slope (m) = (y2 - y1) / (x2 - x1)
Using the coordinates, we get:
m = (-2 - 6) / (4 - 2)
= -8 / 2
= -4
The slope of the right bisector will be the negative reciprocal of the line segment slope. Therefore, the slope of the right bisector is:
Slope of the right bisector = -1 / m
= -1 / -4
= 1/4
Now that we have the slope and the midpoint, we can use the point-slope form of the equation of a line, which is given by:
y - y1 = m(x - x1)
Using the midpoint (3, 2), and the slope 1/4, we get:
y - 2 = 1/4(x - 3)
Simplifying the equation, we get the equation of the right bisector:
4y - 8 = x - 3
x - 4y = -5
To draw the diagram, plot the points E(2, 6) and F(4, -2) on a coordinate plane. Join the two points to form a line segment. Then, draw a line perpendicular to the line segment passing through the midpoint (3, 2).
2. To determine whether two lines are perpendicular, we need to find the slopes of both lines and check if their product is -1.
Given points D(2, 5), E(-2, -3), and F(4, -6), let's find the slopes of line DE and line EF.
Slope of line DE:
m1 = (y2 - y1) / (x2 - x1)
= (-3 - 5) / (-2 - 2)
= -8 / -4
= 2
Slope of line EF:
m2 = (y2 - y1) / (x2 - x1)
= (-6 - (-3)) / (4 - (-2))
= -3 / 6
= -1/2
Now, check if the product of the slopes is -1:
m1 * m2 = 2 * (-1/2) = -1
Since the product is -1, the lines through D and E are perpendicular to the line through E and F.
3. To find the equations of the lines in the triangle formed by the points L(-3, 6), N(3, 2), and P(1, -8), we can use various methods depending on the specific line.
a) Median from N: A median passes through a vertex of a triangle and the midpoint of the opposite side. The midpoint of LP can be found using the midpoint formula:
Midpoint of LP = [(xL + xP) / 2, (yL + yP) / 2]
Using the coordinates, we get:
Midpoint of LP = [(-3 + 1) / 2, (6 + (-8)) / 2]
= [-1, -1]
Now we can find the equation of the line passing through N and the midpoint of LP using the two-point form of a line:
y - y1 = (y2 - y1) / (x2 - x1) * (x - x1)
Using the coordinates (3, 2) for N and (-1, -1) for the midpoint of LP, we get:
y - 2 = (2 - (-1)) / (3 - (-1)) * (x - 3)
Simplifying the equation:
y - 2 = 3/4 * (x - 3)
4y - 8 = 3x - 9
3x - 4y = 1
b) Right bisector of LP: We can use the same method as the first question to determine the equation of the right bisector of LP.
- Find the midpoint of LP using the coordinates (-3, 6) and (1, -8).
- Find the slope of the line segment LP.
- Take the negative reciprocal of the slope of the line segment LP to get the slope of the right bisector.
- Use the point-slope form of the equation of a line with the midpoint and the slope.
c) Altitude from N: An altitude is a line segment drawn from a vertex that is perpendicular to the opposite side. We can use the same method as question 2 to determine if the line through E and F is perpendicular to the line through L and P. If they are perpendicular, we can use the slope of line LP to find the slope of the altitude and then use the point-slope form of the equation to find the equation of the altitude.