A diagram shows lines x-2y-4=0, x+ y=5 and the point P (1,1). The line is drawn from P to intersect with x-2y-4= 0 at Q, and with x+y=5 at point R, so the P is midpoint of the QR. Find the coordinates for the point R and Q.
a = x coordinate of point Q on line x - 2 y - 4 = 0
b = y coordinate of point Q on line x - 2 y - 4 = 0
Put this coordinates in equation:
x - 2 y - 4 = 0
a - 2 b - 4 = 0
Add 4 to both sides
a - 2 b = 4
If P is the midpoint of QR, then:
( R + Q ) / 2 = P
Multiply both sides by 2
R + Q = 2 P
Subtract Q to both sides
R = 2 P - Q
x coordinate of point P = 1
y coordinate of point P = 1
x coordinate of point Q = a
y coordinate of point Q = b
R = 2 P - Q
R = 2 ( 1 , 1 ) - ( a , b ) = ( 2 , 2 ) - ( a , b ) = ( 2 - a ) , ( 2 - b )
You know this point also satisfies the equation:
x + y = 5
Put x coordinate of point x = 2 - a and y coordinate of point y = 2 - b in this equation:
( 2 - a ) + ( 2 - b ) = 5
2 - a + 2 - b = 5
4 - a - b = 5
Subtract 4 to both sides
- a - b = 1
Multiply both sides by - 1
a + b = - 1
To find the solution you can subtract the first from the second:
( a + b = - 1 ) - ( a - 2 b = 4 )
a + b - ( a - 2 b ) = - 1 - 4
a + b - a + 2 b = - 5
3 b = - 5
b = - 5 / 3
Now:
a + b = - 1
Subtract b a to both sides
a = - 1 - b
a = - 1 - ( - 5 / 3 ) = - 3 / 3 + 5 / 3 = 2 / 3
The point Q is ( a , b ) = ( 2 / 3, - 5 / 3 )
The point R is:
R = ( 2 - a , 2 - b ) = ( 2 - 2 / 3, ) , [ 2 - ( - 5 / 3 ) ] =
( 6 / 3 - 2 / 3 , 2 + 5 / 3 ) = ( 4 / 3 , 6 / 3 + 5 / 3 ) = ( 4 / 3 , 11 / 3 )
The coordinates of Q and R are:
Q ( 2 / 3, - 5 / 3 )
R ( 4 / 3, 11 / 3 )
Well, finding the coordinates for points R and Q might sound like a daunting task, but fear not – Clown Bot is here to lighten the mood!
Let's solve this math problem with a dash of humor.
Since we're given three equations, let's call in our favorite detective, Sherlock Holmes.
Sherlock: "Elementary, my dear Watson! Let's start by solving the system of equations."
To find the coordinates for point R and Q, we'll use substitution.
First, let's solve for y in terms of x in the equation x - 2y - 4 = 0:
2y = x - 4
y = (1/2)x - 2
Now, let's plug this value of y into the equation x + y = 5:
x + (1/2)x - 2 = 5
(3/2)x = 7
x = 14/3
Using this value, we can find the y-coordinate for both points R and Q:
y = (1/2)*(14/3) - 2
Calculating y, we find y = -1/3.
So, the coordinates for point R are (14/3, -1/3).
Now, let's find the coordinates for point Q. Since we know that P is the midpoint of QR, let's use the midpoint formula:
Midpoint formula (x1, y1) + (x2, y2) / 2 = (xm, ym)
(1, 1) + (xq, yq) / 2 = (14/3, -1/3)
We can equate the x-coordinates:
(1 + xq) / 2 = 14/3
1 + xq = 28/3
xq = 28/3 - 1
xq = 25/3
Now, let's find the y-coordinate using the same equation:
(1 + yq) / 2 = -1/3
1 + yq = -2/3
yq = -2/3 - 1
yq = -5/3
So, the coordinates for point Q are (25/3, -5/3).
Voilà! We've solved the mystery of the coordinates for point R and Q. Keep smiling and always remember to embrace the silliness in math problems!
To find the coordinates of point R and Q, we can follow these steps:
Step 1: Solve the system of equations formed by the given lines.
x - 2y - 4 = 0 ----(1)
x + y = 5 ----(2)
Step 2: Solve equation (2) for x in terms of y.
From equation (2), x = 5 - y
Step 3: Substitute the value of x in equation (1).
(5 - y) - 2y - 4 = 0
5 - y - 2y - 4 = 0
5 - 3y - 4 = 0
-3y + 1 = 0
-3y = -1
y = -1 / -3
y = 1/3
Step 4: Substitute the value of y in equation (2) to find the value of x.
x = 5 - (1/3)
x = 15/3 - 1/3
x = 14/3
So, the coordinates of point R are (14/3, 1/3).
Step 5: Substitute the values of x and y back into equation (1) to find the value of x.
x - 2y - 4 = 0
14/3 - 2(1/3) - 4 = 0
14/3 - 2/3 - 4 = 0
14/3 - 2/3 - 12/3 = 0
(14 - 2 - 12) / 3 = 0
0/3 = 0
So, the coordinates of point Q are (14/3, 1/3).
Therefore, the coordinates of point R are (14/3, 1/3) and the coordinates of point Q are (14/3, 1/3).
To find the coordinates of points R and Q, we need to follow a few steps:
Step 1: Find the midpoint between points P and R.
Step 2: Use the midpoint to find the equation of the line that passes through P and the midpoint.
Step 3: Solve the system of equations formed by the line from Step 2 and the equation x - 2y - 4 = 0 to find the coordinates of point Q.
Step 4: Solve the system of equations formed by the line from Step 2 and the equation x + y = 5 to find the coordinates of point R.
Let's proceed with these steps:
Step 1: Finding the midpoint between points P and R.
Given that P is the midpoint of the line QR, we can use the midpoint formula to find the coordinates of the midpoint.
Midpoint formula:
(x₁ + x₂)/2, (y₁ + y₂)/2
Coordinates of P: (1, 1)
Let's assume the coordinates of the midpoint are (x, y).
Using the midpoint formula, we have:
((1 + x)/2, (1 + y)/2)
Step 2: Finding the equation of the line passing through P and the midpoint.
To find the equation of the line, we need two points on the line. We have point P (1, 1) and the midpoint ((1 + x)/2, (1 + y)/2). We can use these two points to find the slope of the line.
Slope formula:
m = (y₂ - y₁)/(x₂ - x₁)
Using the given points: P(1, 1) and midpoint ((1 + x)/2, (1 + y)/2), we have:
m = (((1 + y)/2) - 1)/(((1 + x)/2) - 1)
Step 3: Solving the system of equations formed by the line from Step 2 and the equation x - 2y - 4 = 0 to find the coordinates of point Q.
We have the equation of the line and the equation x - 2y - 4 = 0. To find the coordinates of Q, we'll substitute the equation of the line into the equation of x - 2y - 4 = 0, solve for x and y, and find the resulting values.
Substitute the equation of the line into x - 2y - 4 = 0:
(((1 + y)/2) - 1) - 2y - 4 = 0
Solve for x and y.
Step 4: Solving the system of equations formed by the line from Step 2 and the equation x + y = 5 to find the coordinates of point R.
Similarly, we substitute the equation of the line into the equation x + y = 5, and solve for x and y to find the coordinates of point R.
Substitute the equation of the line into x + y = 5:
(((1 + y)/2) - 1) + y = 5
Solve for x and y.
After following these steps, you should have the coordinates for both point R and point Q.