A triangle is formed from the points L(-3,6), N(3,2) and P(1,-8). Find the equation of the following lines:
a) the median from N
b) the right bisector of LP
c) the altitude from N
I'll do a)
you show me what you have done for the other two.
so you want the line from N to the midpoint of LP
midpoint of LP is Q( (1-3)/2, (-8+6)/2) = Q(-1,-1)
slope of NQ = (2-(-1))/(3-(-1) ) = 3/4
equation for NQ:
y-2 = (3/4)(x-3)
4y - 8 = 3x - 9
3x - 4y = 1
or
y = (3/4)x - 1/4
(a) the midpoint M of LP is (-1,-1)
The slope of PM is 4/3
y-2 = 4/3 (x-3)
(b) the slope of the perpendicular bisector of LP (through M) is -3/4, so
y+1 = -3/4 (x+1)
(c) recall that the distance from (h,k) to the line Ax+By+C = 0 is
|Ah+Bk+C|/√(A^2+B^2)
So figure the equation of the line LP and plug it in.
oops. go with Reiny on (a) and thus modify (b)
To find the equation of the lines requested, we need to follow a few steps. Let's go through each question one by one:
a) To find the equation of the median from point N, we need to find the midpoint of the opposite side LP, and then find the equation of the line passing through N and the midpoint.
1. Find the coordinates of the midpoint M of LP:
The formula for finding the midpoint between two points (x1, y1) and (x2, y2) is:
Midpoint M = ((x1 + x2) / 2, (y1 + y2) / 2)
For LP, the coordinates are:
L(-3, 6) and P(1, -8)
Midpoint M = ((-3 + 1) / 2, (6 + (-8)) / 2)
= (-1 / 2, -1)
2. Now that we have the midpoint M(-1/2, -1), we can find the equation of the line passing through N(3, 2) and M(-1/2, -1). We will use the point-slope form of a line: y - y1 = m(x - x1), where m is the slope and (x1, y1) is a point on the line.
Find the slope (m) using the formula: m = (y2 - y1) / (x2 - x1)
m = (2 - (-1)) / (3 - (-1/2))
= 3 / (3 1/2)
= 3 / (7/2)
= 6/7
Using the slope-intercept form of a line (y = mx + b), substitute N(3, 2) for (x1, y1) and use the slope (m) we found:
y - 2 = (6/7)(x - 3)
Simplify and rearrange the equation:
y - 2 = (6/7)x - (18/7)
y = (6/7)x + (4/7)
The equation of the median from N is y = (6/7)x + (4/7).
b) To find the equation of the right bisector of LP, we need to find the midpoint of LP and then calculate the negative reciprocal of the slope of LP to find the slope of the perpendicular bisector.
1. Find the coordinates of the midpoint M of LP.
We already found this in the previous question, which is M(-1/2, -1).
2. Calculate the slope of LP.
Using the formula for slope: m = (y2 - y1) / (x2 - x1)
m = (-8 - 6) / (1 - (-3))
= -14 / 4
= -7 / 2
3. Calculate the negative reciprocal of the slope of LP.
The negative reciprocal is found by changing the sign and flipping the fraction: (2/7)
4. Use the point-slope form of a line with slope (2/7) and point M(-1/2, -1):
y - (-1) = (2/7)(x - (-1/2))
Simplify and rearrange the equation:
y + 1 = (2/7)x + 1/7
y = (2/7)x + 1/7 - 1
y = (2/7)x - 6/7
The equation of the right bisector of LP is y = (2/7)x - 6/7.
c) To find the equation of the altitude from point N, we need to find the slope of LP and calculate the negative reciprocal to find the slope of the altitude. Then, using N(3, 2) as a point, we can determine the equation.
1. Calculate the slope of LP (we already calculated it in question b).
m = -7/2
2. Calculate the negative reciprocal of the slope of LP.
The negative reciprocal is found by changing the sign and flipping the fraction: (2/7)
3. Use the point-slope form of a line with slope (2/7) and point N(3, 2):
y - 2 = (2/7)(x - 3)
Simplify and rearrange the equation:
y - 2 = (2/7)x - 6/7
y = (2/7)x - 6/7 + 14/7
y = (2/7)x + 8/7
The equation of the altitude from N is y = (2/7)x + 8/7.
That's it! We've found the equations of the lines requested.