Final answer:
To calculate the distance between two points on a line with a given slope, we use the slope formula to first find the x-coordinate of one point, and then apply the distance formula. In this instance, the distance is approximately 3 times the slope of 2, thus the correct answer is Option 2: 3·.
Step-by-step explanation:
To find the distance between two points with a given slope, we can use the distance formula combined with the slope formula. The slope is defined as the change in y over the change in x. Given that the slope of the line joining the points (x, 5) and (1, 2) is 2, we can first use the slope formula to find x.
The slope (·) between two points (x1, y1) and (x2, y2) is given by:
· = (y2 - y1) / (x2 - x1)
Plugging the values we get:
2 = (2 - 5) / (1 - x)
2 = -3 / (1 - x)
To find x, solve for x:
-3 = 2 * (1 - x)
-3 = 2 - 2x
2x = 2 + 3
x = 5 / 2
x = 2.5
Now we have both points (2.5, 5) and (1, 2), and we can calculate the distance using the distance formula:
Distance = √((x2 - x1)² + (y2 - y1)²)
Distance = √((1 - 2.5)² + (2 - 5)²)
Distance = √((-1.5)² + (-3)²)
Distance = √(2.25 + 9)
Distance = √(11.25)
Distance = 3.354 (approximately 3 times the slope 2)
Therefore, the distance between the points is approximately 3 times the slope, making Option 2: 3· the correct choice.