The points P(-2,2) , Q(4,4) and R(5,2) are vertices of a Triangle. The perpendicular bisector of PQ and the line through P parallel to QR intersect at S.
Find the coordinates of S.
you have the points
calculate the slopes and find the equations of the two lines
find the intersection by solving the system of two equations
I didn't understand
Can u please solve then forward me the solutions
Show the solutions
To find the coordinates of S, we need to find the equations of the perpendicular bisector of PQ and the line through P parallel to QR. Then, we can find the intersection point of these two lines.
Step 1: Find the coordinates of the midpoint of PQ.
The midpoint of PQ is the average of the x-coordinates and the average of the y-coordinates of points P and Q.
Midpoint coordinates = ((-2 + 4) / 2, (2 + 4) / 2) = (1, 3)
Step 2: Find the slope of PQ.
The slope of a line passing through two points (x₁, y₁) and (x₂, y₂) is given by the formula:
m = (y₂ - y₁) / (x₂ - x₁)
Slope of PQ = (4 - 2) / (4 - (-2)) = 2 / 6 = 1 / 3
Step 3: Find the negative reciprocal of the slope of PQ.
The negative reciprocal of a slope is obtained by taking the negative reciprocal of the fraction.
Negative reciprocal of PQ's slope = -1 / (1 / 3) = -3
Step 4: Find the equation of the perpendicular bisector of PQ using the midpoint and the negative reciprocal of the slope.
The equation of a line passing through a point (x₁, y₁) with slope m is given by the point-slope form:
(y - y₁) = m(x - x₁)
Using the midpoint (1, 3) and the negative reciprocal slope -3:
(y - 3) = -3(x - 1)
Simplify the equation:
y - 3 = -3x + 3
y = -3x + 6
Step 5: Find the equation of the line passing through P parallel to QR.
Since the line is parallel to QR, it has the same slope as QR. The slope of QR is given by the formula:
m = (y₂ - y₁) / (x₂ - x₁)
Slope of QR = (2 - 4) / (5 - (-2)) = -2 / 7
Using the point-slope form with point P(-2, 2) and slope -2/7:
(y - 2) = (-2/7)(x - (-2))
Simplify the equation:
(y - 2) = (-2/7)(x + 2)
(y - 2) = (-2/7)x - 4/7
y = (-2/7)x + 18/7
Step 6: Find the intersection point of the perpendicular bisector and the parallel line.
To find the coordinates of the intersection point, we need to solve the system of equations formed by the equations of the perpendicular bisector and the line through P parallel to QR.
Substitute y in the equation of the perpendicular bisector with the equation of the parallel line:
-3x + 6 = (-2/7)x + 18/7
Now, solve for x:
Multiply through by 7 to clear the fraction:
-21x + 42 = -2x + 18
Combine like terms:
-21x + 2x = 18 - 42
-19x = -24
Divide by -19 to solve for x:
x = (-24) / (-19) = 24/19
Substitute the value of x back into either equation to solve for y:
y = (-2/7)(24/19) + 18/7 = -3.15
Therefore, the coordinates of point S are approximately (24/19, -3.15).