a straight line passes through p(-1,1) and q(3,4)
a) find the length of line pq
b) find the equation of the perpendicular bisector of the line pq leaving it in the form y=mx+c
c) determine the equation of line to line pq and passes through point (2,3) leaving your answer in double intercept form. hence state the y intercept
a) The length of line PQ can be found using the distance formula:
distance = sqrt((x2 - x1)^2 + (y2 - y1)^2)
distance = sqrt((3 - (-1))^2 + (4 - 1)^2)
distance = sqrt(4^2 + 3^2)
distance = sqrt(16 + 9)
distance = sqrt(25)
distance = 5
Therefore, the length of line PQ is 5 units.
b) To find the equation of the perpendicular bisector of line PQ, we need to find the midpoint of the line segment PQ and the slope of the line PQ.
Midpoint coordinates:
x = (x1 + x2)/2 = (-1 + 3)/2 = 1/2
y = (y1 + y2)/2 = (1 + 4)/2 = 5/2
Midpoint coordinates: (1/2, 5/2)
Slope of line PQ:
m = (y2 - y1)/(x2 - x1) = (4 - 1)/(3 - (-1)) = 3/4
To find the slope of the perpendicular bisector, we take the negative reciprocal of the slope of line PQ:
m_perpendicular = -1/m = -1/(3/4) = -4/3
Using the midpoint coordinates and the slope of the perpendicular bisector, we can write the equation in slope-intercept form:
y - y1 = m_perpendicular(x - x1)
y - 5/2 = -4/3(x - 1/2)
y - 5/2 = -4/3x + 2/3
y = -4/3x + 2/3 + 5/2
y = -4/3x + 4/6 + 15/6
y = -4/3x + 19/6
Therefore, the equation of the perpendicular bisector of line PQ is y = -4/3x + 19/6.
c) To find the equation of the line passing through point (2,3) and intersecting line PQ, we need to find the slope of the line PQ and use the point-slope form.
Slope of line PQ:
m = (y2 - y1)/(x2 - x1) = (4 - 1)/(3 - (-1)) = 3/4
Using the point-slope form:
y - y1 = m(x - x1)
y - 3 = (3/4)(x - 2)
y - 3 = (3/4)x - 3/2
y = (3/4)x - 3/2 + 3
y = (3/4)x - 3/2 + 6/2
y = (3/4)x + 3/2
Therefore, the equation of the line passing through point (2,3) and intersecting line PQ is y = (3/4)x + 3/2.
The double intercept form of the equation can be found by setting y = 0 in the equation and solving for x:
0 = (3/4)x + 3/2
(3/4)x = -3/2
x = (-3/2)/(3/4)
x = -3/2 * 4/3
x = -2
Hence, the y-intercept is -2.
To calculate the length of the line segment PQ, we will use the distance formula.
a) Length of line PQ:
The distance formula is given by:
d = √((x2 - x1)^2 + (y2 - y1)^2)
Substituting the given coordinates, the length of PQ can be calculated as:
d = √((3 - (-1))^2 + (4 - 1)^2)
= √((3 + 1)^2 + (4 - 1)^2)
= √(4^2 + 3^2)
= √(16 + 9)
= √25
= 5
Therefore, the length of line PQ is 5 units.
b) Equation of the perpendicular bisector:
To find the equation of the perpendicular bisector of PQ, we need to find the midpoint of the line segment PQ, then determine the slope of PQ, and finally find the negative reciprocal of that slope.
1. Finding the midpoint of line PQ:
The midpoint formula is given by:
(x, y) = ((x1 + x2)/2, (y1 + y2)/2)
Substituting the given coordinates, the midpoint of PQ can be calculated as:
(x, y) = ((-1 + 3)/2, (1 + 4)/2)
= (2/2, 5/2)
= (1, 2.5)
So, the midpoint of PQ is (1, 2.5).
2. Finding the slope of line PQ:
The slope formula is given by:
m = (y2 - y1)/(x2 - x1)
Substituting the given coordinates, the slope of PQ can be calculated as:
m = (4 - 1)/(3 - (-1))
= 3/4
3. Finding the negative reciprocal slope:
The negative reciprocal of 3/4 is -4/3.
Therefore, the slope of the perpendicular bisector is -4/3.
Finally, to determine the equation of the perpendicular bisector in the form y = mx + c, we substitute the midpoint coordinates and the slope into the equation:
y = mx + c
2.5 = (-4/3)(1) + c
2.5 = -4/3 + c
c = 2.5 + (4/3)
c = 7/3
Hence, the equation of the perpendicular bisector of line PQ is:
y = (-4/3)x + 7/3.
c) Equation of line through point (2,3) and perpendicular to PQ:
Since the slope of the perpendicular bisector is -4/3, the slope of the line through point (2,3) and perpendicular to PQ will be the negative reciprocal of -4/3, which is 3/4.
Using the point-slope form of a line, the equation of the line through point (2,3) and perpendicular to PQ can be written as:
y - y1 = m(x - x1)
y - 3 = (3/4)(x - 2)
To find the y-intercept, we substitute the coordinates of (2,3) into the equation and simplify:
y - 3 = (3/4)(2 - 2)
y - 3 = 0
y = 3
Therefore, the y-intercept is 3.
Hence, the equation of the line through point (2,3) and perpendicular to PQ, in double-intercept form, is:
4y - 12 = 3x - 6, or 3x - 4y + 6 = 0.
The y-intercept is 3.
To find the answers to the given questions, we can follow these steps:
a) Finding the length of line PQ:
We can use the distance formula to find the length of line PQ, which is the distance between the points P(-1, 1) and Q(3, 4):
Using the distance formula:
d = sqrt((x2 - x1)^2 + (y2 - y1)^2)
Substituting the values:
d = sqrt((3 - (-1))^2 + (4 - 1)^2)
= sqrt((4)^2 + (3)^2)
= sqrt(16 + 9)
= sqrt(25)
= 5
Therefore, the length of line PQ is 5 units.
b) Finding the equation of the perpendicular bisector of line PQ:
To find the equation of the perpendicular bisector, we need to find the midpoint of line PQ and the slope of line PQ.
Step 1: Finding the midpoint of line PQ:
The midpoint (M) can be found using the formula:
M = ((x1 + x2)/2, (y1 + y2)/2)
Substituting the values:
M = ((-1 + 3)/2, (1 + 4)/2)
= (1/2, 5/2)
So the midpoint of line PQ is M(1/2, 5/2).
Step 2: Finding the slope of line PQ:
The slope (m) of a line passing through two points can be found using the formula:
m = (y2 - y1)/(x2 - x1)
Substituting the values:
m = (4 - 1)/(3 - (-1))
= 3/4
So the slope of line PQ is 3/4.
Step 3: Finding the slope of the perpendicular bisector:
The perpendicular bisector has a slope that is the negative reciprocal of the slope of line PQ. Therefore, the slope of the perpendicular bisector (m_perpendicular) will be -1/m.
m_perpendicular = -1/(m)
= -1/(3/4)
= -4/3
Step 4: Writing the equation in the form y = mx + c:
Since M(1/2, 5/2) is a point on the perpendicular bisector, we can substitute the values of the midpoint and the slope into the point-slope form of a line equation:
(y - y1) = m(x - x1)
Substituting the values:
(y - 5/2) = (-4/3)(x - 1/2)
Expanding and rearranging:
3y - 15/2 = -4x + 2/3
Multiplying all terms by 6 to eliminate fractions:
6(3y - 15/2) = 6(-4x + 2/3)
18y - 45 = -24x + 4
Rearranging the equation:
24x + 18y = 49
Thus, the equation of the perpendicular bisector of line PQ is 24x + 18y = 49.
c) Finding the equation of the line passing through point (2, 3) and perpendicular to line PQ:
Since this line is perpendicular to line PQ, it will have a slope that is the negative reciprocal of the slope of PQ. We already found the slope of line PQ to be 3/4, so the slope of this new line will be -4/3.
Using the point-slope form of a line equation:
(y - y1) = m(x - x1)
Substituting the values:
(y - 3) = (-4/3)(x - 2)
Expanding and rearranging:
3y - 9 = -4x + 8/3
Multiplying all terms by 3 to eliminate fractions:
9(3y - 9) = 9(-4x + 8/3)
27y - 81 = -36x + 8
Rearranging the equation:
36x + 27y = 89
Thus, the equation of the line passing through point (2, 3) and perpendicular to line PQ is 36x + 27y = 89.
To determine the y-intercept, we can set x = 0 and solve for y:
36(0) + 27y = 89
27y = 89
y = 89/27
Therefore, the y-intercept is y = 89/27.
(a) √((3+1)^2 + (4-1)^2) = √(4^2+3^2) = √25 = 5
(b) the midpoint of PQ is (P+Q)/2 = ((-1+3)/2,(1+4)/2) = (1,5/2)
The slope of PQ is (4-1)/(3+1) = 3/4
so the slope of the perpendicular is -4/3
So, the point-slope equation is y - 5/2 = -4/3 (x-1)
(c) Not sure what you mean by "line to line pq"
There are many lines that intersect pq. But once you figure out what that means, recall that if a line has
x-intercept = (a,0)
y-intercept = (0,b)
the intercept form of the line is
x/a + y/b = 1