Unit 1 Lesson 1b: The Distance Formula

Lesson Overview and Metadata

Unit 1 Lesson 1b (designated as code U1L1b) covers the topic of The Distance Formula. The student work is recorded by Rummel Pilamonta for Period 3 on the date 9/15.

Finding the length of a line segment joining two points in the coordinate plane is identical to determining the distance between those two points. The fundamental foundation for calculating distance on a coordinate plane is the Pythagorean Theorem.

Fundamentals of Calculating Distance

There are two primary mathematical options available for evaluating the length of a segment on a coordinate plane. Both methods share the exact same geometric principles.

Option 1: The Pythagorean Theorem

The Pythagorean Theorem applies to right triangles and relates the lengths of the two legs, denoted as $a$ and $b$, to the length of the hypotenuse, denoted as $c$.

The algebraic formula for the Pythagorean Theorem is given by:

$a^2 + b^2 = c^2$

In a right triangle diagram, side $a$ represents one leg, side $b$ represents the adjacent perpendicular leg, and side $c$ represents the hypotenuse opposite the right angle.

Option 2: The Distance Formula

The Distance Formula provides a direct algebraic method for calculating distance using coordinate points.

The Distance Formula is written as:

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

In this formula, $(x_1, y_1)$ and $(x_2, y_2)$ represent the coordinates of the endpoints of the line segment. This formula is specifically used to calculate the total length $d$ of a segment on the coordinate grid.

The Distance Formula is simply the Pythagorean Theorem explicitly adapted to coordinate geometry. It remains a standard mathematical tool used across geometry, algebra, and advanced calculus courses.

Standard Procedure for Finding Distance

To find the distance between any two coordinate points using either the Distance Formula or the Pythagorean Theorem, follow a standardized four-step procedure:

Step 1: State the formula ($a^2 + b^2 = c^2$ or $d = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2}$).

Step 2: Label the given points explicitly as $(x_1, y_1)$ and $(x_2, y_2)$.

Step 3: Substitute the known coordinate values into the selected formula.

Step 4: Evaluate the expression algebraically to determine the final distance value.

Worked Example: Distance Between Two Points

Exercise #1 requires finding the distance between the points $(-3, 1)$ and $(5, 7)$ using both the Pythagorean Theorem and the Distance Formula.

Solution via Method 1: Pythagorean Theorem

First, plot the points $(-3, 1)$ and $(5, 7)$ on the coordinate grid and form a right triangle by drawing horizontal and vertical line segments connecting the two endpoints.

Calculate the horizontal leg length $a$: moving along the x-axis from $x = -3$ to $x = 5$ gives a horizontal distance of $|5 - (-3)| = 8$ units.

Calculate the vertical leg length $b$: moving along the y-axis from $y = 1$ to $y = 7$ gives a vertical distance of $|7 - 1| = 6$ units.

Apply the Pythagorean Theorem formula $a^2 + b^2 = c^2$:

$8^2 + 6^2 = c^2$

$64 + 36 = c^2$

$100 = c^2$

$\sqrt{100} = \sqrt{c^2}$

$10 = c$

The hypotenuse length, and therefore the segment distance, is $10$ units.

Solution via Method 2: Distance Formula

Label the coordinates as $(x_1, y_1) = (-3, 1)$ and $(x_2, y_2) = (5, 7)$.

State the Distance Formula:

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

Substitute the labeled coordinate values into the formula:

$d = \sqrt{(5 - (-3))^2 + (7 - 1)^2}$

Simplify the terms within the parentheses:

$d = \sqrt{(8)^2 + (6)^2}$

Evaluate the squares of both quantities:

$d = \sqrt{64 + 36}$

Sum the values under the radical sign:

$d = \sqrt{100}$

Compute the principal square root:

$d = 10$

Both computational options yield the identical result of $10$ units for the distance between $(-3, 1)$ and $(5, 7)$.

Conceptual Questions and Analysis

Question 1: Geometric Representation of Formula Terms

What do the $(x_2 - x_1)$ and $(y_2 - y_1)$ parts of the Distance Formula represent on a right triangle?

The term $(x_2 - x_1)$ represents the horizontal distance along the x-axis, which corresponds to the length of the horizontal leg of the right triangle. The term $(y_2 - y_1)$ represents the vertical distance along the y-axis, corresponding to the length of the vertical leg of the right triangle.

Question 2: Necessity of the Square Root Step

Why must the final step in both methods involve taking a square root?

The Pythagorean Theorem computes the square of the hypotenuse, $c^2 = a^2 + b^2$. To isolate the variable $c$ (or distance $d$) and find the actual linear distance rather than the squared value, a square root must be applied to both sides of the equation, yielding $c = \sqrt{a^2 + b^2}$.

Here are two practice problems based on your notes for you to try:

Problem 1:
Find the distance between the points $(-2, 4)$ and $(4, 12)$ using both the Pythagorean Theorem and the Distance Formula.

Problem 2:
Find the distance between the points $(1, -3)$ and $(7, 5)$.

Try following the 4 steps from your notes:

1. State the formula
2. Label the given points
3. Substitute the values
4. Evaluate algebraically

Share your answers when you are ready!