Beast Academy 5D (age 13) — Pythagorean Paths: Using the Pythagorean Theorem to Find Distance Between Two Points

What you need to know

If you have two points on a coordinate plane, say (x₁, y₁) and (x₂, y₂), you can find the distance between them by making a right triangle and using the Pythagorean Theorem.

Derivation (short)

Make a right triangle whose legs are the horizontal and vertical differences between the points:
horizontal leg = |x₂ − x₁|, vertical leg = |y₂ − y₁|. By Pythagoras, distance d satisfies:

d = sqrt((x₂ − x₁)² + (y₂ − y₁)²)

Step-by-step recipe

1. Label the points (x₁, y₁) and (x₂, y₂).
2. Compute dx = x₂ − x₁ and dy = y₂ − y₁. (Order doesn't matter because you square them.)
3. Compute dx² + dy².
4. Take the square root: d = sqrt(dx² + dy²). Simplify the radical if possible.

Worked examples

Example 1: A(1, 2) and B(5, 5).

1. dx = 5 − 1 = 4, dy = 5 − 2 = 3.
2. dx² + dy² = 4² + 3² = 16 + 9 = 25.
3. d = sqrt(25) = 5.

So AB = 5 (this is a 3–4–5 right triangle).

Example 2: C(−2, 4) and D(3, −1).

1. dx = 3 − (−2) = 5, dy = −1 − 4 = −5.
2. dx² + dy² = 5² + (−5)² = 25 + 25 = 50.
3. d = sqrt(50) = sqrt(25·2) = 5·sqrt(2).

Example 3 (vertical or horizontal): E(2, 7) and F(2, 1).

dx = 0, dy = −6 → d = sqrt(0 + 36) = 6. When points share the same x or y, just take the difference in the other coordinate.

Tips and shortcuts

• If dx and dy form a Pythagorean triple (like 3 and 4, or 5 and 12), the distance will be an integer.
• You don’t need to keep the absolute value signs because squaring removes signs: (x₂ − x₁)² = (x₁ − x₂)².
• Count squares on the grid if the points are easy to see on graph paper — that helps you set dx and dy quickly.
• Always simplify radicals (e.g., sqrt(50) → 5·sqrt(2)).

Common mistakes to watch for

• Forgetting to subtract x’s for the horizontal leg and y’s for the vertical leg (mixing coordinates up).
• Using dx and dy in the wrong order for something else — order doesn’t matter here, but you must use x difference for the horizontal leg and y difference for the vertical leg.
• Not simplifying radicals or assuming sqrt(50) = 7 (estimate vs exact).

Practice advice for Beast Academy problems (pg. 58–60, #142–155)

- Do #142–150 first (required). These will practice finding dx and dy, squaring, and simplifying. Work each problem by drawing the right triangle on the grid, labeling dx and dy, and using the formula.
- #151–155 are optional — try them after you finish the earlier ones for extra challenge (they may include larger coordinates, negative coordinates, or require simplification of radicals).
- Check your work by estimating the distance (count squares or use a Pythagorean triple) to see if your exact answer is reasonable.

Quick checklist before you finish a problem

1. Are the coordinates labeled correctly (x first, y second)?
2. Did I compute dx and dy correctly?
3. Did I square then add before taking the square root?
4. Is my final answer simplified?

If you want, send one of the practice problems here (like #142 or #147) and I’ll walk through it step-by-step with you.