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

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a) Well, to determine if point C(63,54) is on the branch pipeline, we'll need to take our detective hats and magnifying glasses out. If the branch pipeline is truly a straight line, we just need to check if point C lies on that line. So, let's calculate the slope of the main pipeline and see if it applies to point C as well. If the slope checks out, then it's a positive confirmation, but if it doesn't, then C is probably just trying to be a poser and not really on the branch pipeline.

The slope of the main pipeline is calculated using the formula: slope = (change in y / change in x). So, for the main pipeline, we have:

slope = (40 - 60) / (65 - 45)
= -20 / 20
= -1

Now, let's calculate the slope for the segment between point A and C:

slope = (54 - 60) / (63 - 45)
= -6 / 18
= -1/3

Since the slope of the segment between A and C is not equal to the slope of the main pipeline, it seems that C is not on the branch pipeline. Sorry, C! Close, but no pipeline connection for you this time.

b) Ah, the shortest route conundrum. Well, if you want to connect point C(63,54) to the main pipeline, the optimal way would be to find the perpendicular bisector of the segment between A(45,60) and B(65,40). This magical line will intersect the main pipeline at just the right spot.

To find the midpoint of the segment between A and B, we can use the formula:

midpoint = ( (x1 + x2) / 2, (y1 + y2) / 2)
midpoint = ( (45 + 65) / 2, (60 + 40) / 2)
midpoint = (55, 50)

Now that we have the midpoint, we can calculate the slope of the main pipeline and find its negative reciprocal to obtain the slope of the perpendicular bisector. Once we have the slope of the perpendicular bisector, we can use point C(63,54) to calculate the equation of the line.

But hey, don't sweat it. The easiest way is probably just to connect C to the main pipeline with a straight line. It might not be the shortest route, but it surely saves you the headache of all this math.

a) To determine if point C(63,54) is on the branch pipeline, we can use the concept of slope. The main pipeline runs straight from point A(45,60) to point B(65,40). We can calculate the slope of the main pipeline using the formula:

slope = (y2 - y1) / (x2 - x1)

Let's calculate the slope of the main pipeline:

slope_main = (40 - 60) / (65 - 45)
= -20 / 20
= -1

Now, let's calculate the slope between point A and point C:

slope_AC = (54 - 60) / (63 - 45)
= -6 / 18
= -1/3

Since the slope of the main pipeline (-1) and the slope between points A and C (-1/3) are different, point C(63,54) is not on the branch pipeline.

b) To find the shortest route for connecting point C(63,54) to the main pipeline, we can use the concept of perpendicular lines. The shortest route would be a perpendicular line from point C to the main pipeline.

To find the perpendicular line, we need to find the negative reciprocal of the slope of the main pipeline. The negative reciprocal is obtained by flipping the fraction and changing the sign.

The slope of the main pipeline is -1, so the negative reciprocal is 1.

Now, we need to find the equation of the perpendicular line passing through point C(63,54). We can use the point-slope form of a line:

y - y1 = m(x - x1)

Plugging in the values, we get:

y - 54 = 1(x - 63)
y - 54 = x - 63
y = x - 63 + 54
y = x - 9

Therefore, the equation of the shortest route for connecting point C(63,54) to the main pipeline is y = x - 9.