Grade 8

Intermediate

8 min

Lesson 9 of 12

Distance on the Coordinate Plane

Not Started

Use the Pythagorean Theorem to find the distance between any two points.

If you were designing a video game, how would the computer know exactly how far an arrow travels to hit a target? It doesn't use a ruler—it uses a 2,500-year-old mathematical secret that turns any diagonal line into a simple puzzle.

Learning Objectives

1.Apply the Pythagorean Theorem to find the distance between any two points on a coordinate grid.
2.Construct a right triangle on a plane to visualize horizontal and vertical change.
3.Solve real-world distance problems using the Distance Formula.

The Hidden Triangle

When we look at two points on a coordinate plane, the direct path between them is often a diagonal line. On a grid, we can easily count horizontal or vertical units, but we can't 'count' diagonally. To solve this, we imagine the diagonal line as the hypotenuse (the longest side) of a right triangle. By drawing a horizontal line from one point and a vertical line from the other, they meet at a 90-degree angle. This allows us to use the Pythagorean Theorem:

a^2 + b^2 = c^2

, where

a

and

b

are the lengths of the legs we just drew, and

c

is the distance we want to find.

Find the distance between Point A $(1, 1)$ and Point B $(4, 5)$ .

1. Find the horizontal distance (leg $a$ ): $4 - 1 = 3$ units. 2. Find the vertical distance (leg $b$ ): $5 - 1 = 4$ units. 3. Plug into the theorem: $3^2 + 4^2 = c^2$ . 4. Calculate: $9 + 16 = 25$ . 5. Solve for $c$ : $\sqrt{25} = 5$ .

The distance is 5 units.

Quick Check

If the horizontal distance between two points is 6 and the vertical distance is 8, what is the direct distance between them?

Calculate

6^2 + 8^2

and then take the square root.

Answer

10 units

The Distance Formula Shortcut

What if the points are far apart or involve negative numbers? We can generalize our triangle method into the Distance Formula. Instead of drawing a triangle every time, we use the coordinates

(x_1, y_1)

and

(x_2, y_2)

directly. The horizontal leg is the difference in

x

values

(x_2 - x_1)

, and the vertical leg is the difference in

y

values

(y_2 - y_1)

. When we square these differences, the result is always positive, which is why the direction doesn't matter! The formula is:

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

Find the distance between $P(-2, 3)$ and $Q(4, -1)$ .

1. Identify coordinates: $x_1 = -2, y_1 = 3, x_2 = 4, y_2 = -1$ . 2. Subtract $x$ values: $4 - (-2) = 6$ . Square it: $6^2 = 36$ . 3. Subtract $y$ values: $-1 - 3 = -4$ . Square it: $(-4)^2 = 16$ . 4. Add them: $36 + 16 = 52$ . 5. Find the square root: $d = \sqrt{52} \approx 7.21$ units.

Quick Check

Why does the Distance Formula square the differences of the coordinates?

Answer

To ensure the side lengths are positive (since distance cannot be negative) and to follow the $a^2 + b^2$ part of the Pythagorean Theorem.

Real-World Geometry

In the real world, coordinates help us find perimeters of shapes or the shortest path between locations. If you have a triangle with vertices on a map, you can find the length of all three sides using the distance formula. This is how GPS software calculates the 'as the crow flies' distance between your current location and your destination. Remember: the shortest distance between two points is always a straight line, which represents the hypotenuse of the change in their coordinates.

A park is shaped like a triangle with corners at $(0,0)$ , $(0,3)$ , and $(4,0)$ . What is the perimeter of the park?

1. Side 1 (Vertical): From $(0,0)$ to $(0,3)$ is 3 units. 2. Side 2 (Horizontal): From $(0,0)$ to $(4,0)$ is 4 units. 3. Side 3 (Diagonal): Use the formula $\sqrt{3^2 + 4^2} = \sqrt{25} = 5$ units. 4. Total Perimeter: $3 + 4 + 5 = 12$ units.

Key Takeaways

1The distance between two points is the hypotenuse of a right triangle formed by the horizontal and vertical distances.
2The Distance Formula is $d = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2}$ .
3Squaring the differences in coordinates removes negative signs, making the order of subtraction flexible.
4Always take the square root at the final step to find the actual distance.

Quick Check

Multiple Choice

What is the distance between $(0,0)$ and $(6,8)$ ?

Multiple Choice

Which part of a right triangle does the distance between two points represent?

True/False

The distance between $(2, 5)$ and $(2, 9)$ can be found without the Pythagorean Theorem.

What's Next?

Review Tomorrow

Tomorrow morning, try to write down the Distance Formula from memory and explain how it relates to $a^2 + b^2 = c^2$ .

Practice Activity

Open a digital map or graph paper, pick two random points, and calculate the distance between them. Verify it by drawing the triangle and counting the squares!

Browse

Browse

Distance on the Coordinate Plane

Learning Objectives

The Hidden Triangle

The Distance Formula Shortcut

Real-World Geometry

Key Takeaways

Quick Check

What's Next?

Distance on the Coordinate Plane

Learning Objectives

The Hidden Triangle

The Distance Formula Shortcut

Real-World Geometry

Key Takeaways

Quick Check

What's Next?