A triangle is formed from the points L(-3, 6), N(3, 2) and P(1, -8). Find the equation of the following lines.
Find:
the median from N
the right bisector of LP
the altitude from N
We can find the equation of a straight line if we know the slope and a point on it, so each part of your question must be reduced to that requirement.
Median: a line from a vertex to the midpoint of the opposite side
So we need the midpoint of LP, which is ( (-3+1)/2 , (6-8)/2 ) = (-1,-1)
Using that point and the point N(3,2) we can find the slope
slope of median = (-1-2)/((-1-3) = 3/4
I now have the slope and a point, so we can find the equation.
Whatever method you learned:
y-2 = (3/4)(x-3)
4y-8 = 3x - 9
3x - 4y = 1
right bisector of AP
you need the midpoint of AP, that's your usable point of the equation
find the slope of AP
the right-bisector's slope would be the reciprocal of the slope of AP
using those hints .....
altitude from N
so you use point N as your point.
the altitude must meet the line AP at right angles, so it is perpendicular.
So ....
Let me know what the other two equations are after you found them.
To find the equations of the lines in question, we can use various formulas and concepts from coordinate geometry. Let's solve each part step by step:
1. Finding the median from N:
The median of a triangle is a line segment that connects a vertex to the midpoint of the opposite side. In this case, we need to find the midpoint of the side LP and then determine the equation of the line passing through N and that midpoint.
Step 1: Find the midpoint of LP.
To find the midpoint, we can use the midpoint formula:
Midpoint = ((x1 + x2)/2, (y1 + y2)/2)
For points L(-3, 6) and P(1, -8):
Midpoint = ((-3 + 1)/2, (6 + (-8))/2)
= (-2/2, -2/2)
= (-1, -1)
Step 2: Find the equation of the line passing through N(-3, 6) and the midpoint (-1, -1).
We can use the two-point form of the equation of a line, which is:
y - y1 = (y2 - y1)/(x2 - x1) * (x - x1)
For the given points N(-3, 6) and midpoint (-1, -1):
y - 6 = (-1 - 6)/(-1 - (-3)) * (x - (-3))
y - 6 = (-7)/(2) * (x + 3)
y - 6 = (-7/2)(x + 3)
2y - 12 = -7x - 21
7x + 2y + 9 = 0
Therefore, the equation of the median from N is 7x + 2y + 9 = 0.
2. Finding the right bisector of LP:
The right bisector of a line segment is a line that is perpendicular to the segment and passes through its midpoint. To find the equation of the right bisector of LP, we need to find the midpoint and the slope of LP and then determine the equation of the line perpendicular to LP passing through the midpoint.
Step 1: Find the midpoint of LP.
We found the midpoint of LP in the previous step, which is (-1, -1).
Step 2: Find the slope of LP.
The slope of a line passing through two points (x1, y1) and (x2, y2) can be calculated using the formula:
Slope = (y2 - y1)/(x2 - x1)
For points L(-3, 6) and P(1, -8):
Slope = (-8 - 6)/(1 - (-3))
= (-14)/(4)
= -7/2
Step 3: Find the negative reciprocal of the slope of LP. This will give us the slope of the right bisector line.
The negative reciprocal of -7/2 is 2/7.
Step 4: Use the point-slope form of the equation of a line to find the equation of the right bisector.
The point-slope form of the equation of a line is:
y - y1 = m(x - x1)
Using the midpoint (-1, -1) and the slope 2/7:
y - (-1) = (2/7)(x - (-1))
y + 1 = (2/7)(x + 1)
7y + 7 = 2x + 2
2x - 7y + 5 = 0
Therefore, the equation of the right bisector of LP is 2x - 7y + 5 = 0.
3. Finding the altitude from N:
The altitude is a line segment from a vertex of the triangle that is perpendicular to the opposite side. To find the equation of the altitude from N, we need to find the slope of LP and then determine the equation of the line passing through N with the negative reciprocal slope.
Step 1: Find the slope of LP.
We found the slope of LP in the previous step, which is -7/2.
Step 2: Find the negative reciprocal of the slope of LP. This will give us the slope of the altitude line.
The negative reciprocal of -7/2 is 2/7.
Step 3: Use the point-slope form of the equation of a line to find the equation of the altitude.
Using the point N(-3, 6) and the slope 2/7:
y - 6 = (2/7)(x - (-3))
y - 6 = (2/7)(x + 3)
7y - 42 = 2x + 6
2x - 7y + 36 = 0
Therefore, the equation of the altitude from N is 2x - 7y + 36 = 0.