The distance formula is derived from the Pythagorean Theorem. What is the distance between points A and B? We can use the Pythagorean Theorem to find the distance. Using an example of two points and then expanding to a more generalized situation, students can see how the distance formula can be determined. What is the distance between points A and B?
Overhead 1
One way to determine the distance from point A to point B is to use the Pythagorean Theorem. The point A is at (6, 4) and the point B is at (-3, -4). Drawing lines to create a right triangle gives us:
Overhead 2
With the creation of the right triangle, the length of the hypotenuse (*AB*) is the distance from point A to point B. The length of *AC* is the difference in the y-coordinates of the points, or 4 – (-4), which is 8. The length of *BC* is the difference in the x-coordinates of the points, or 6 – (-3), or 9. Using the Pythagorean Theorem, we get:
a^{2} + b^{2} = c^{2}
9^{2} + 8^{2} = c^{2}
81 + 64 = c^{2}
145 = c^{2}
so that c ≈ 12.04
So that the length of *AB* is 12.04, meaning that the distance from A to B is about 12.04 units.
We can now generalize this process for any two points. We can generalize the points A and B to be at any coordinate, for example A can be at (*x*_{1},*y*_{1}) and B can be (*x*_{2},*y*_{2}).
Overhead 3
Since the points are identified in generic terms, we can see the legs of the right angle as changes in the arbitrary coordinates, or *y*_{2} – *y*_{1} for the vertical length and *x*_{2} – *x*_{1}.
Overhead 4
Now, using the Pythagorean Theorem,
(*x*_{2} – *x*_{1})^{2} + (*y*_{2} – *y*_{1})^{2} = (distance)^{2}
Now, solving for distance by taking the square root of both sides,
distance = √(*x*_{2} – *x*_{1})^{2} + (*y*_{2} – *y*_{1})^{2}
which is the distance formula for any two points in the coordinate plane.
Looking back at the earlier example, with point A at (6, 4) and point B at (-3, -4), then,
distance = √(–3–6)^{2} + (–4–4)^{2} = √(–9)^{2} + (–8)^{2} = √81 + 64 = √145 ≈ 12.04
Note that the distance is the same no matter the derivation.
In this first example, it is clear that the Pythagorean Theorem and the distance formula tell us only limited information, the distance between the two points. We can gain additional information from these points, including the slope between the points. Using the same points, the slope is
This indicates that the direction between these two points follows a line with a slope of and is a way that students can see how the same idea, the difference of x and y coordinates can be used in different ways to find different characteristics of two points. This connection between distance and direction can preview the study of vectors, which give both direction and distance.
Worksheet: HSA Practice 1
Worksheet: HSA Practice 2
**Follow-Up Activity**
Discuss the worksheet with the students in the next lesson. Focus on student explanations of which process they feel is best, not necessarily the selection.
**Supplemental Activities**
Please use the HSA public release items to share problems like these with your students. |