A map shows a main gas pipeline running straight from A(45,60) to B(65,40)
a) Is the point C(63,54) on the branch pipeline? Explain your reasoning
b) What is the shortest route for connecting point C to the main pipeline.Explain
**OMGGGG PLZZZ HELPPP, THANK YOUUUUUU!!!!!!!!!!!
dfr
a) To determine if point C(63,54) is on the branch pipeline, we need to check if it lies on the line segment AB.
The equation of the line passing through points A(45,60) and B(65,40) can be found using the slope-intercept form:
y = mx + b
where m is the slope and b is the y-intercept.
The slope (m) can be calculated using the formula:
m = (y2 - y1) / (x2 - x1)
Using the coordinates A(45,60) and B(65,40), the slope (m) is:
m = (40 - 60) / (65 - 45)
m = -20 / 20
m = -1
Now, we can find the y-intercept (b) using the point-slope form:
y - y1 = m(x - x1)
Using point A(45,60), we have:
y - 60 = -1(x - 45)
y - 60 = -x + 45
y = -x + 45 + 60
y = -x + 105
Now, we can substitute the coordinates of point C(63,54) into the equation and check if it satisfies the equation.
54 = -63 + 105
54 = 42
Since 54 is not equal to 42, point C(63,54) does not lie on the branch pipeline.
b) The shortest route for connecting point C(63,54) to the main pipeline is a perpendicular line drawn from point C to the line segment AB. This perpendicular line will intersect with the line segment AB at a right angle.
To find the equation of the perpendicular line, we need to find the negative reciprocal of the slope of the line segment AB.
The slope of AB (m) was calculated earlier as -1. The negative reciprocal of -1 is 1. Therefore, the slope of the perpendicular line is 1.
Now we can use the point-slope form to find the equation of the perpendicular line passing through point C(63,54):
y - y1 = m(xx - x1)
Using point C(63,54) and the slope m = 1:
y - 54 = 1(x - 63)
y - 54 = x - 63
y = x - 63 + 54
y = x - 9
This equation represents the perpendicular line passing through point C(63,54). The point of intersection of this line with the line segment AB will be the shortest route for connecting point C to the main pipeline.
a) To determine if point C(63,54) is on the branch pipeline, we can calculate the equation of the line that connects points A and B using the formula for the slope of a line:
Slope (m) = (change in y) / (change in x)
m = (40-60) / (65-45)
m = -20 / 20
m = -1
Using the slope-intercept form of a linear equation (y = mx + b), we can substitute the coordinates of point A (45,60) and solve for b:
60 = -1 * 45 + b
b = 60 + 45
b = 105
So, the equation of the line that connects points A and B is: y = -x + 105.
Now, we need to check if point C(63,54) satisfies this equation. Substituting the x and y values of point C into the equation:
54 = -63 + 105
54 = 42
Since 54 is not equal to 42, point C does not lie on the branch pipeline.
b) To determine the shortest route for connecting point C(63,54) to the main pipeline, we need to find the perpendicular distance from point C to the line that connects points A and B.
The shortest route will be when the perpendicular from point C intersects the line connecting A and B at a right angle. We can calculate the equation of the line perpendicular to the line connecting A and B and passing through point C using the negative reciprocal of the slope (-1/m).
The slope of the line connecting A and B is -1. Therefore, the slope of the line perpendicular to it will be 1.
Using the point-slope form of a linear equation (y - y1 = m(x - x1)), we can substitute the coordinates of point C (63,54) and solve for the equation of the line perpendicular to AB:
y - 54 = (1)(x - 63)
y = x - 9
Now we need to find the intersection point of the two lines: y = -x + 105 and y = x - 9. By solving these equations simultaneously, we can find the coordinates of the intersection point, which will give us the shortest route:
-x + 105 = x - 9
2x = 114
x = 57
Substituting the value of x back into either equation, we can find the y-coordinate:
y = 57 - 9
y = 48
Therefore, the intersection point of the two lines is point D(57,48), which represents the shortest route for connecting point C to the main pipeline.
The equation of a line passing two given points P1(x1,y1) and P2(x2,y2) is:
(y-y1)/(y2-y1) = (x-x1)/(x2-x1)
Substitute the coordinates of the two given points and simplify to find the formula of the line, L1.
(hint: The equation I get is Ax+By+C=0 where C is -105)
a.
Substitute the value of x of the third point (P3) into L1 and see if the corresponding value of y equals the y-coordinate of P3.
b.
The distance of the point P3(x3,y3) from a line Ax+By+C=0 is given by
|Ax+By+C|/sqrt(A²+B²)
I get about 8.5.