Find the bisectors of the interior angles of the triangle whose sides are the line 7x+y-7=0, x+y+1=0 and x+7y-4=0.
7x+y-7=0 intersects x+y+1=0 at (4/3,-7/3).
7x+y-7=0 makes an angle of 98.13°
x+y+1=0 makes an angle of 135°
with the x-axis.
So, the bisector makes an angle of 116.56°, with slope -2
So, the equation of the line is
y + 7/3 = -2(x - 4/3)
3y+7 = -2(3x-4)
6x+3y+15=0
Now work the other pairs of lines.
To find the bisectors of the interior angles of the triangle, we first need to find the equations of the three sides of the triangle.
Given the equations of the lines that form the sides of the triangle:
1) 7x + y - 7 = 0
2) x + y + 1 = 0
3) x + 7y - 4 = 0
To find the bisectors of the interior angles, we need to find the equations of the perpendicular bisectors to each side.
Step 1: Find the slopes of the given lines.
A line in the form of y = mx + c, where m is the slope, and c is the y-intercept. The slope (m) can be found by rearranging each equation into this form.
1) 7x + y - 7 = 0
Rearrange to: y = -7x + 7
The slope of this line is -7.
2) x + y + 1 = 0
Rearrange to: y = -x - 1
The slope of this line is -1.
3) x + 7y - 4 = 0
Rearrange to: 7y = -x + 4
y = -1/7 * x + 4/7
The slope of this line is -1/7.
Step 2: Find the midpoints of each side.
The midpoint of a line segment can be found by taking the average of the x-coordinates and the y-coordinates of the endpoints.
For line 1: y = -7x + 7
The slope is -7.
Since there is no y-intercept, we can set y = 0 to find the x-coordinate.
0 = -7x + 7
7x = 7
x = 1
Substituting the value of x back into the equation, we find the y-coordinate:
y = -7(1) + 7
y = 7 - 7
y = 0
The midpoint of line 1 is (1, 0).
For line 2: y = -x - 1
The slope is -1.
Again, we can set y = 0 and solve for x:
0 = -x - 1
x = -1
The y-coordinate is:
y = -( -1) - 1
y = 1 - 1
y = 0
The midpoint of line 2 is (-1, 0).
For line 3: y = -1/7 * x + 4/7
The slope is -1/7.
Setting y = 0 to find the x-coordinate:
0 = -1/7 * x + 4/7
1/7 * x = 4/7
x = 4
The y-coordinate is:
y = -1/7 * (4) + 4/7
y = -4/7 + 4/7
y = 0
The midpoint of line 3 is (4, 0).
Step 3: Find the slopes of the perpendicular bisectors.
The slopes of the perpendicular bisectors will be the negative reciprocals of the slopes of the lines.
For line 1, the slope is -7, so the slope of the perpendicular bisector will be 1/7.
For line 2, the slope is -1, so the slope of the perpendicular bisector will be 1.
For line 3, the slope is -1/7, so the slope of the perpendicular bisector will be 7.
Step 4: Write the equations of the perpendicular bisectors.
Using the slope-intercept form y = mx + c, where m is the slope and c is the y-intercept, we can substitute the midpoint coordinates and the respective slopes to obtain the equations of the perpendicular bisectors.
For line 1, with midpoint (1, 0) and slope 1/7:
y = (1/7)x + c
Substituting the midpoint coordinates:
0 = (1/7)(1) + c
0 = 1/7 + c
c = -1/7
The equation of the perpendicular bisector for line 1 is y = (1/7)x - 1/7.
For line 2, with midpoint (-1, 0) and slope 1:
y = 1x + c
Substituting the midpoint coordinates:
0 = 1(-1) + c
0 = -1 + c
c = 1
The equation of the perpendicular bisector for line 2 is y = x + 1.
For line 3, with midpoint (4, 0) and slope 7:
y = 7x + c
Substituting the midpoint coordinates:
0 = 7(4) + c
0 = 28 + c
c = -28
The equation of the perpendicular bisector for line 3 is y = 7x - 28.
Therefore, the bisectors of the interior angles of the triangle formed by the given lines are:
1) y = (1/7)x - 1/7
2) y = x + 1
3) y = 7x - 28