A line segment has endpoints j ( 2, 4 ) and l ( 6, 8 ). The point k is the midpoint of jl. What is an equation of a line perpendicular to jl and passing through k?
y=-x+10
Slope of segment, a
= (y2-y1)/(x2-x1)
= (8-4)/(6-2)=1
Mid-point
=( (x1+x2)/2, (y1+y2)/2 )
=( (2+6)/2, (4+8)/2 )
= M(4,6)
Slope of line perpendicular to segment,a1
= -1/a
= -1/(1)
=-1
Equation of perpendicular line:
(y-y1)=a1(x-x1)
y-6 = -1(x-4)
or
y=-x + 10
Check that it goes through M
-4+10=6 ok.
Slope of segment, a
= (y2-y1)/(x2-x1)
= (8-4)/(6-2)=1
Mid-point
=( (x1+x2)/2, (y1+y2)/2 )
=( (2+6)/2, (4+8)/2 )
= M(4,6)
Slope of line perpendicular to segment,a1
= -1/a
= -1/(1)
=-1
Equation of perpendicular line:
(y-y1)=a1(x-x1)
y-6 = -1(x-4)
or
y=-x + 10
Check that it goes through M
-4+10=6 ok.
y=x+2
Why did the line segment go to therapy? Because it had "endpoints" issues! Now, let's solve your problem.
First, we need to find the midpoint of jl, which is the average of the x-coordinates and the average of the y-coordinates. So, the x-coordinate of point k is (2+6)/2 = 4, and the y-coordinate is (4+8)/2 = 6. Therefore, the coordinates of point k are (4, 6).
Now, we need to find the slope of jl. The slope (m) of a line passing through two points (x1, y1) and (x2, y2) is given by the formula m = (y2 - y1)/(x2 - x1). For jl, the slope is (8 - 4)/(6 - 2) = 4/4 = 1.
Since we want to find a line perpendicular to jl, we need to find the negative reciprocal of the slope. The negative reciprocal of 1 is -1/1 = -1.
Now, we can use the slope-intercept form of a line (y = mx + b), where m is the slope and b is the y-intercept.
With point k (4, 6), we can substitute these values into the equation and solve for b:
6 = -1(4) + b,
6 = -4 + b,
b = 10.
Therefore, the equation of the line perpendicular to jl and passing through point k is y = -x + 10.
I hope that helps, and remember to always keep your lines perpendicular and your jokes perpendicular too!
To find an equation of a line perpendicular to JL and passing through K, we need to determine the slope of JL and then use the negative reciprocal of that slope as the slope of the perpendicular line.
The slope of JL can be found using the slope formula: m = (y2 - y1) / (x2 - x1).
For JL, the coordinates of point J are (2, 4) and the coordinates of point L are (6, 8). Applying the formula, we have:
m = (8 - 4) / (6 - 2)
m = 4 / 4
m = 1
Therefore, the slope of JL is 1.
Next, we need to find the coordinates of the midpoint K. The midpoint formula is given by ( (x1 + x2) / 2, (y1 + y2) / 2 ).
For JL, we have:
x1 = 2, x2 = 6
y1 = 4, y2 = 8
Using the midpoint formula, we can calculate the coordinates of point K:
xK = (2 + 6) / 2 = 8 / 2 = 4
yK = (4 + 8) / 2 = 12 / 2 = 6
Therefore, the coordinates of K are (4, 6).
Now that we have the slope of JL (1) and the coordinates of K (4, 6), we can find the slope of the perpendicular line by taking the negative reciprocal of 1.
The negative reciprocal of 1 is -1/1, which simplifies to -1.
So, the slope of the perpendicular line is -1.
Finally, we can use the point-slope form of a line to find the equation of the line perpendicular to JL and passing through K. The point-slope form is given by y - y1 = m(x - x1), where (x1, y1) represents a point on the line and m is the slope.
Plugging in the value of K (4, 6) and the slope (-1), we have:
y - 6 = -1(x - 4)
Simplifying, we get:
y - 6 = -x + 4
Rearranging, we obtain the equation of the line:
y = -x + 10
So, the equation of the line perpendicular to JL and passing through K is y = -x + 10.