# Calculations Using Points on a Graph

Hello and welcome to this video about calculations using points on the coordinate plane!

So first off, let’s remember that a one-dimensional number line is a representation of all real numbers that extends infinitely in both the positive and negative directions and looks something like this: When two number lines intersect at a right angle at their 0 coordinates, the two-dimensional coordinate plane is formed, which typically looks something like this: Generally, the horizontal axis is called the $$x$$-axis and the vertical axis is called the $$y$$-axis. The point where the axes intersect is called the origin. Let’s plot a point on the coordinate plane. The location of Point A is given as a horizontal component ($$x$$) and a vertical component (y) and written as $$(x,y)$$. From the origin, to get to Point A, we would count two units to the right and three units up. So, the coordinates of Point A are $$(2,3)$$. Likewise, the coordinates of the origin are $$(0,0)$$.

Since the coordinate plane is comprised of number lines, they can be viewed “zoomed in” or “zoomed out” as much as necessary to convey data or a story. For example, the $$x$$-axis here has been “zoomed in”: Because of the scale of the $$x$$-axis, the coordinate points of Point B are $$(\frac{1}{2},3)$$.

And the $$x$$-axis has been zoomed in here, while the $$y$$-axis has been zoomed out: Because of the scales of the axes, the coordinates of Point C are $$(0.2,300)$$.

Using the coordinates of points on a coordinate plane, we can calculate the distance between two points.

The distance formula (an application of the Pythagorean theorem) looks like this:

$$D=\sqrt{(x_2-x_1)^2+(y_2-y_1)^2}$$

In words, it’s “the square root of the horizontal distance squared plus the vertical distance squared.” In this case, the formula can be used, but since Points A and D lie on the same horizontal gridline, all we need to do is count squares (this also works for two points on the same vertical gridline). The distance from Point A to Point D is 6 because the two points are 6 squares apart.

Using the formula yields the same result:

$$D=\sqrt{(x_2-x_1)^2+(y_2-y_1)^2}$$

$$D=\sqrt{(2-(-4))^2+(3-3)^2}$$

$$D=\sqrt{(6)^2+(0)^2}$$

$$D=\sqrt{36+0}$$

$$D=\sqrt{36}$$

$$D=6$$ In this case, we can see that the coordinates of Point B are $$(\frac{1}{2},3)$$ and the coordinates of Point E are $$(\frac{3}{2},-6)$$. In order to determine the distance between the two points, we’ll need to use the formula because the points do not share a gridline, so we can’t simply count squares to determine the distance.

$$D=\sqrt{(x_2-x_1)^2+(y_2-y_1)^2}$$

$$D=\sqrt{(\frac{3}{2}-\frac{1}{2})^2+(-6-3)^2}$$

$$D=\sqrt{((1)^2+(-9)^2}$$

$$D=\sqrt{1+81}$$

$$D=\sqrt{82}$$

$$D≈9.055$$ In this case, we’re going to need to estimate the coordinates of Point F because it doesn’t lie at the intersection of two gridlines. It is often the case that we need to estimate the coordinates of a point. Let’s estimate Point F to be located at $$(-0.4,-350)$$. Now, let’s use the distance formula to get a good idea of the distance between Points C and F:

$$D=\sqrt{(x_2-x_1)^2+(y_2-y_1)^2}$$

$$D=\sqrt{(-0.4-0.2)^2+(-350-300)^2}$$

$$D=\sqrt{(-0.6)^2+(-650)^2}$$

$$D=\sqrt{0.36+422,500}$$

$$D=\sqrt{422,500.36}$$

$$D≈650$$

Using the coordinates of points on a coordinate plane, we can also calculate the coordinates of a point that lies exactly halfway between two points, which is known as the midpoint.

The formula for finding the midpoint coordinates looks like this:

$$M=(\frac{x_1+x_2}{2},\frac{y_1+y_2}{2})$$

In words, it’s easy to remember as “the average of the $$x$$’s, the average of the $$y$$’s.” As we saw previously, the distance between Points A and D is 6. Half of that distance is 3. Since Points A and D lie on the same gridline, the midpoint will lie there as well, at the coordinates $$(-1,3)$$.

We can verify this with the formula:

$$M=(\frac{x_1+x_2}{2},\frac{y_1+y_2}{2})$$

$$M=(\frac{-4+2}{2},\frac{3+3}{2})$$

$$M=(\frac{-2}{2},\frac{6}{2})$$

$$M=(-1,3)$$ Even though we can always calculate half of the distance between two points, that won’t tell us the coordinates of the midpoint in this example like the formula does:

$$M=(\frac{x_1+x_2}{2},\frac{y_1+y_2}{2})$$

$$M=(\frac{\frac{1}{2}+\frac{3}{2}}{2},\frac{3+(-6)}{2})$$

$$M=(\frac{2}{2},\frac{-3}{2})$$

$$M=(1,-\frac{3}{2})$$

To graph, we’ll need to estimate the location of $$-\frac{3}{2}$$. Previously, we estimated the coordinates of Point F to be $$(-0.4,-350)$$. Let’s use the formula to estimate the midpoint between Points C and F:

$$M=(\frac{x_1+x_2}{2},\frac{y_1+y_2}{2})$$

$$M≈(\frac{0.2+(-0.4)}{2},\frac{-350+300}{2})$$

$$M≈(\frac{-0.2}{2},\frac{-50}{2})$$

$$M≈(-0.1,-25)$$

To graph, we’ll need to estimate the location of -25.

I hope that this video helped you understand how to perform calculations using the points on a coordinate plane! Thanks for watching, and happy studying!

## Practice Questions

Question #1:

What is the distance between the points (2, 7) and (31, 25)?

22.54

34.13

27.92

31.79

The correct answer is 34.13. The Distance Formula is:
$$D=\sqrt{(x_2-x_1)^2+(y_2-y_1)^2}$$
Plugging in these points results in:
$$D=\sqrt{(31-2)^2+(25-7)^2}$$
$$=\sqrt{(29)^2+(18)^2}$$
$$=\sqrt{841+324}$$
$$=\sqrt{1,165}≈34.13$$

Question #2:

What is the midpoint of the points (3, 5) and (4, 17)?

($$\frac{7}{2}$$, 11)

(7, 22)

($$\frac{5}{2}$$, 6)

(5, 12)

The correct answer is ($$\frac{7}{2}$$, 11). The formula for finding midpoint is:
$$M=(\frac{x_1+x_2}{2},\frac{y_1+y_2}{2})$$
Plugging in these two points results in:
$$(\frac{3+4}{2},\frac{5+17}{2})=(\frac{7}{2},\frac{22}{2})=(\frac{7}{2},11)$$

Question #3:

What is the distance between the points (4, 7) and (7, 4)?

6.81

2.76

7.13

4.24

The correct answer is 4.24. The Distance Formula is:
$$D=\sqrt{(x_2-x_1)^2+(y_2-y_1)^2}$$
Plugging in these points results in:
$$D=\sqrt{(7-4)^2+(4-7)^2}$$
$$=\sqrt{(3)^2+(-3)^2}$$
$$=\sqrt{9+9}$$
$$=\sqrt{18}≈4.24$$

Question #4:

What is the midpoint of the points (-7, 3) and (14, -9)?

(-11, 8)

(7, -6)

($$-\frac{11}{2}$$, 4)

($$\frac{7}{2}$$, -3)

The correct answer is ($$\frac{7}{2}$$, -3). The formula for finding midpoint is:
$$M=(\frac{x_1+x_2}{2},\frac{y_1+y_2}{2})$$
Plugging in these two points results in:
$$(\frac{-7+14}{2},\frac{3+(-9)}{2})=(\frac{7}{2},\frac{-6}{2})=(\frac{7}{2},-3)$$

Question #5:

What is the distance between the points (-12, 4) and (6, -9)?

22.20

14.19

27.63

19.41

The correct answer is 22.20. The Distance Formula is:
$$D=\sqrt{(x_2-x_1)^2+(y_2-y_1)^2}$$
Plugging in these points results in:
$$D=\sqrt{(6-(-12))^2+(-9-4)^2}$$
$$=\sqrt{(18)^2+(-13)^2}$$
$$=\sqrt{324+169}$$
$$=\sqrt{493}≈22.20$$

Question #6:

What is the distance between the points $$(-3,7)$$ and $$(16,11)$$?

318.73

19.42

21

377

The correct answer is 19.42. The formula for distance between two points, $$(x_1, y_1)$$ and $$(x_2, y_2)$$, is:
$$D=\sqrt{(x_2-x_1)^2+(y_2-y_1)^2}$$

Plug in the given points and solve.
$$D=\sqrt{(16-(-3))^2+(11-7)^2}=\sqrt{(19)^2+(4)^2}=\sqrt{361+16}$$
$$=\sqrt{377}≈19.42$$

Question #7:

What is the distance between the points $$(14, -9)$$ and $$(12, 11)$$?

20.1

404

13

412.7

The correct answer is 20.1. The formula for distance between two points, $$(x_1, y_1)$$ and $$(x_2, y_2)$$, is:
$$D=\sqrt{(x_2-x_1)^2+(y_2-y_1)^2}$$

Plug in the given points and solve.
$$D=\sqrt{(12-14)^2+(11-(-9))^2}=\sqrt{(-2)^2+(20)^2}=\sqrt{4+400}$$
$$=\sqrt{404}≈20.1$$

Question #8:

What is the distance between the points $$(-7, -1)$$ and $$(8, -2)$$?

226

211.72

19

15.03

The correct answer is 15.03. The formula for distance between two points, $$(x_1, y_1)$$ and $$(x_2, y_2)$$, is:
$$D=\sqrt{(x_2-x_1)^2+(y_2-y_1)^2}$$

Plug in the given points and solve.
$$D=\sqrt{(8-(-7))^2+(-2-(-1))^2}=\sqrt{(15)^2+(-1)^2}=\sqrt{225+1}$$
$$=\sqrt{226}≈15.03$$

Question #9:

What is the distance between the points $$(21, -7)$$ and $$(-13, 12)$$?

38.95

1,493.86

42

1,517

The correct answer is 38.95. The formula for distance between two points, $$(x_1, y_1)$$ and $$(x_2, y_2)$$, is:
$$D=\sqrt{(x_2-x_1)^2+(y_2-y_1)^2}$$

Plug in the given points and solve.
$$D=\sqrt{(-13-21)^2+(12-(-7))^2}=\sqrt{(-34)^2+(19)^2}=\sqrt{1{,}156+361}$$
$$=\sqrt{1{,}517}≈38.95$$

Question #10:

What is the distance between the points $$(16, 17)$$ and $$(-14, 2)$$?

1,125

1,197.63

33.54

36

The correct answer is 33.54. The formula for distance between two points, $$(x_1, y_1)$$ and $$(x_2, y_2)$$, is:
$$D=\sqrt{(x_2-x_1)^2+(y_2-y_1)^2}$$

Plug in the given points and solve.
$$D=\sqrt{(-14-16)^2+(2-17)^2}=\sqrt{(-30)^2+(-15)^2}=\sqrt{900+225}$$
$$=\sqrt{1{,}125}≈33.54$$