A point that is between 50 and 60 units away from (7, -2) is (60, 0), and the distance between these two points is 53.04 units.
Finding a Point Within a Specific Distance Range
Determining a point that falls within a particular distance range from another given point involves understanding the concept of distance in a coordinate plane. There are numerous points that can satisfy such a condition. The key is to find one specific example and verify its distance.
The Solution Point
One such point that meets the criteria is (60, 0). When calculating the distance from (7, -2) to (60, 0), it falls precisely within the specified range of 50 to 60 units.
Understanding Distance in a Coordinate Plane
The distance between any two points $(x_1, y_1)$ and $(x_2, y_2)$ in a two-dimensional coordinate system is calculated using the distance formula, which is derived from the Pythagorean theorem:
$D = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2}$
Let's apply this formula to confirm the distance between (7, -2) and (60, 0):
- Point 1: $(x_1, y_1) = (7, -2)$
- Point 2: $(x_2, y_2) = (60, 0)$
- Calculate the difference in x-coordinates:
$x_2 - x_1 = 60 - 7 = 53$ - Calculate the difference in y-coordinates:
$y_2 - y_1 = 0 - (-2) = 0 + 2 = 2$ - Square these differences:
$(53)^2 = 2809$
$(2)^2 = 4$ - Sum the squared differences:
$2809 + 4 = 2813$ - Take the square root of the sum:
$\sqrt{2813} \approx 53.0377...$
Rounding to two decimal places, the distance is 53.04 units. Since 53.04 is greater than 50 and less than 60, the point (60, 0) successfully fits the criteria.
Summary of Points and Distance
| Point 1 | Point 2 | Calculated Distance | Meets Criteria (50-60 units) |
|---|---|---|---|
| (7, -2) | (60, 0) | 53.04 units | Yes |
