How to Solve: Distance Formula

How to Solve: Distance Formula

(For Students Who Want to Ace Their Exam & Teachers Who Need a Ready-to-Record Script)

Introduction

"Ever wondered how your GPS calculates the exact miles between two cities—or how a pilot knows the shortest flight path? The distance formula is the secret. Master it, and you’ll crush geometry problems on your exam—and in real life!

What You Need To Know First

Before diving into the distance formula, make sure you understand: 1. Coordinate Plane Basics – How to plot points (x, y) and read their coordinates. 2. Pythagorean Theorem – The foundation of the distance formula: a² + b² = c². 3. Square Roots – How to simplify √(a² + b²) to find the final distance.

If any of these feel shaky, pause and review them first!

Key Vocabulary

Formulas To Know

1. Distance Formula

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

Variables: - d = distance between two points - (x₁, y₁) = coordinates of the first point - (x₂, y₂) = coordinates of the second point

Memorise This? ✅ YES! (Not always given on exam sheets.)

Why It Works: The distance formula is just the Pythagorean Theorem in disguise! The differences (x₂ – x₁) and (y₂ – y₁) form the legs of a right triangle, and d is the hypotenuse.

Step-by-Step Method

Follow these steps for EVERY distance problem:

1. Label the points.
2. Write down (x₁, y₁) and (x₂, y₂) clearly.

3. Tip: It doesn’t matter which point is first or second—just be consistent!

4. Plug into the formula.

5. Substitute the coordinates into:
   [ d = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2} ]

6. Calculate the differences.

7. Find (x₂ – x₁) and (y₂ – y₁).

8. Warning: Keep the order the same! (Don’t mix up x and y.)

9. Square the differences.

10. Square both results: (x₂ – x₁)² and (y₂ – y₁)².

11. Add the squares.

12. Add the two squared numbers together.

13. Take the square root.

14. Simplify √(sum) to get the final distance.

15. Tip: If the sum is a perfect square (like 25), the answer is an integer (5).

16. Check units.

17. If the problem gives units (e.g., miles, cm), include them in your answer.

Worked Example Using the Steps

Problem: Find the distance between (1, 2) and (4, 6).

1. Label the points:
2. (x₁, y₁) = (1, 2)

3. (x₂, y₂) = (4, 6)

4. Plug into the formula:
   [ d = \sqrt{(4 - 1)^2 + (6 - 2)^2} ]

5. Calculate the differences:

6. (4 – 1) = 3

7. (6 – 2) = 4

8. Square the differences:

9. 3² = 9

10. 4² = 16

11. Add the squares:

12. 9 + 16 = 25

13. Take the square root:

14. √25 = 5

15. Check units:

16. No units given → Answer: 5 units

Worked Examples

Example 1 – Basic

Problem: Find the distance between (0, 0) and (3, 4).

Solution: 1. (x₁, y₁) = (0, 0), (x₂, y₂) = (3, 4) 2. [ d = \sqrt{(3 - 0)^2 + (4 - 0)^2} ] 3. (3 – 0) = 3, (4 – 0) = 4 4. 3² = 9, 4² = 16 5. 9 + 16 = 25 6. √25 = 5

Answer: 5 units

What we did and why: We used the distance formula to find the hypotenuse of a 3-4-5 right triangle. This is the simplest case—no negative numbers or decimals.

Example 2 – Medium (Negative Coordinates)

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

Solution: 1. (x₁, y₁) = (-2, 5), (x₂, y₂) = (3, -1) 2. [ d = \sqrt{(3 - (-2))^2 + (-1 - 5)^2} ] 3. (3 – (-2)) = 5, (-1 – 5) = -6 4. 5² = 25, (-6)² = 36 5. 25 + 36 = 61 6. √61 ≈ 7.81 (or leave as √61 if exact form is required)

Answer: √61 units (or ≈ 7.81 units)

What we did and why: Negative coordinates make the subtraction trickier. Remember: subtracting a negative is adding (e.g., 3 – (-2) = 5). Squaring removes the negative sign, so the order doesn’t matter.

Example 3 – Exam Style (Disguised Problem)

Problem: A drone flies from point A (1, 7) to point B (6, -5). How far does it travel? Give your answer to 1 decimal place.

Solution: 1. (x₁, y₁) = (1, 7), (x₂, y₂) = (6, -5) 2. [ d = \sqrt{(6 - 1)^2 + (-5 - 7)^2} ] 3. (6 – 1) = 5, (-5 – 7) = -12 4. 5² = 25, (-12)² = 144 5. 25 + 144 = 169 6. √169 = 13.0

Answer: 13.0 units

What we did and why: This is a real exam question—it tests if you can apply the formula under time pressure. The numbers are larger, but the steps are identical. Always check if the answer needs rounding!

Common Mistakes

Exam Traps

1-Minute Recap

"Alright, let’s lock this in for your exam. The distance formula is just the Pythagorean Theorem in coordinate form: take the difference in x’s, square it; take the difference in y’s, square it; add them, and take the square root. Label your points, plug in carefully, and don’t rush the signs—especially with negatives! If the numbers get messy, leave the answer as a square root unless the question asks for a decimal. And always double-check: did you square both differences? Did you add before taking the square root? Nail these steps, and you’ll ace every distance problem on test day. Now go practice—you’ve got this!