Find the distance between these points in two ways: (-2,5) and (4,13).

Question

Find the distance between these points in two ways: (-2,5) and (4,13).

a). Use the point (-2,5) as (x₁,y₁) and the point (4,13) as (x₂,y₂) in the distance formula.

b). Use the point (4,13) as (x₁,y₁) and the point (-2,5) as (x₂,y₂) in the distance formula.

c). If your answers are the same, explain why that happened. If your answers are not the same, look for a mistake in part a or b.

Answer 1

Hi Yuliya! To find the distance between two points, you can use the distance formula, where distance = the square root of ((x₂ - x₁)² + (y₂ - y₁)²).

a) For part a of this problem, start by labeling your first point (-2,5) as (x₁, y₁) so x₁ = -2 and y₁ = 5. This is saying that the x value of our first point is -2 and the y value is 5. Next, label your second point (4,13) as (x₂, y₂), so x₂ = 4 and y₂= 13. When you plug these values into the formula, you'll be finding: distance = the square root of ((4 - (-2))² + (13 - 5)²). If we simplify this, then the distance = the square root of ((4+2)² + (8)²), or, simplified further, the distance equals the square root of (6² + 8²). The square root of (6² + 8²) is the square root of (36+64), or the square root of 100, which is 10. So 10 is the distance between these two points.

b) For part b, you're applying the same formula but you making (4, 13) the first point, so x₁ = 4 and y₁ = 13. (-2, 5) is the second point, so x₂ = -2 and y₂= 5. When you substitute these values into the formula, you get that the distance = the square root of ((-2 - 4)² + (5 - 13)²), which simplifies to the square root of ((-6)² + (-8)²). This further simplifies to the distance = the square root of (36 + 64), or the square root of 100, which is 10.

c) The answers are the same because the physical distance between the points will not change, regardless of which point you choose as the first point, (x₁, y₁) or the second point, (x₂, y₂).

I hope this helps!