Two points in the Cartesian plane have the coordinates 2.00 meters, negative 4.00 meters and negative 3.00 meters, 3.00 meters. Find the distance between them.
Let’s start by drawing a sketch of these two points on the Cartesian plane. On these axes, each tick mark represents a distance of 1.00 meters. The first point mentioned has an 𝑥-value of 2.00 meters then a 𝑦-value of negative 4.00 meters; that’s here on the plane.
The second point is at negative 3.00 meters plus 3.00 meters; that’s here. We’re asked to solve for the distance between these two points. That distance which we’ll call 𝑑 is the length of the straight line path from one point to the other.
If we draw out the change in 𝑥 position from one point to another, calling that 𝛥𝑥, and the change in 𝑦 position between the two points, calling that 𝛥𝑦, we can see that we formed a right triangle and 𝑑 is the hypotenuse of that triangle.
Solving for distance in general between points on a Cartesian plane, that distance 𝑑 is equal to the square root of 𝛥𝑥 squared plus 𝛥𝑦 squared. Looking at the two points that we’ve been given, let’s call the first point 𝑝 sub one and the second point 𝑝 sub two.
If we apply the distance relationship to our two points 𝑝 one and 𝑝 two, then we can rewrite 𝛥𝑥 and 𝛥𝑦 as the difference between the 𝑥- and 𝑦-coordinates of 𝑝 one and 𝑝 two, respectively.
𝛥𝑥 is equal to 2.00 minus minus 3.00 meters or 5.00 meters. 𝛥𝑦 equals negative 4.00 meters minus 3.00 meters or negative 7.00 meters. When we square 𝛥𝑥 and 𝛥𝑦, add them together, and take their square root, we find a total distance of 8.60 meters. That’s how far point one and point two are away from one another.