Beast Academy 5D (Age 14) — Pythagorean Paths: Find Distances on a Coordinate Grid (Pages 58–60, #142–155)

Step-by-step guide for Beast Academy 5D Pythagorean Paths. Learn how to apply the Pythagorean Theorem to find distances between two points on a coordinate grid (8.G.B.8). Includes worked examples, tips, and hints for practice problems #142–155.

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Goal: Use the Pythagorean Theorem to find the distance between two points (x1, y1) and (x2, y2) on a coordinate plane.

Short rule (Distance Formula):

Distance = sqrt((x2 − x1)2 + (y2 − y1)2)

Where this comes from: draw a right triangle whose horizontal leg is the difference in x-coordinates and vertical leg is the difference in y-coordinates. Then apply the Pythagorean Theorem: hypotenuse2 = horizontal2 + vertical2.

Step-by-step method

  1. Label the two points (x1, y1) and (x2, y2).
  2. Find the horizontal leg: Δx = x2 − x1. (Order doesn’t matter because you square it; Δx might be negative but squaring removes the sign.)
  3. Find the vertical leg: Δy = y2 − y1.
  4. Compute Δx2 + Δy2.
  5. Take the square root: distance = sqrt(Δx2 + Δy2).
  6. Simplify the radical if possible (factor out perfect squares). If the result is a whole number, give that; otherwise give a simplified radical or a decimal if the problem asks for it.

Worked examples

Example 1 — integer result (3–4–5 triangle)

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

  1. Δx = 4 − 1 = 3
  2. Δy = 6 − 2 = 4
  3. Distance = sqrt(32 + 42) = sqrt(9 + 16) = sqrt(25) = 5

Example 2 — simplify a radical

Find the distance between (−2, 3) and (4, −1).

  1. Δx = 4 − (−2) = 6
  2. Δy = −1 − 3 = −4 (square removes the sign)
  3. Distance = sqrt(62 + (−4)2) = sqrt(36 + 16) = sqrt(52).
  4. Simplify: sqrt(52) = sqrt(4·13) = 2·sqrt(13).

Example 3 — fractional coordinates

Find the distance between (1/2, 3) and (−1/2, −1).

  1. Δx = (−1/2) − (1/2) = −1 → Δx2 = 1
  2. Δy = −1 − 3 = −4 → Δy2 = 16
  3. Distance = sqrt(1 + 16) = sqrt(17). (Already simplified.)

Tips and common mistakes

  • It doesn’t matter which point you call (x1, y1) and which is (x2, y2) — the result is the same.
  • Don’t forget to square negative differences — squaring removes the minus sign.
  • If Δx or Δy is 0, the distance is just the absolute value of the other difference (horizontal or vertical distance).
  • Always try to simplify radicals: factor out perfect squares (4, 9, 16, …) before leaving your answer as a square root.
  • If the problem uses a grid, draw the right triangle on the grid to see if you get a 3-4-5 or other Pythagorean triple — that makes mental calculation faster.

Hints for the assigned Beast Academy problems (5D pages 58–60, #142–155)

  • #142–150: Most of these will ask you to draw a right triangle between two grid points: compute Δx and Δy and apply the distance formula. Expect some answers to be integers (look for 3-4-5 or 5-12-13 triples) and some to be simplified radicals.
  • #151–155 (optional / trickier): these may include non-integer coordinates, require simplifying radicals carefully, or ask you to compare distances. Work carefully with signs and simplify fully. If asked to compare distances, square both distances and compare the sums Δx2 + Δy2 instead of taking square roots — that avoids extra work.

Example hint for a typical optional problem: if you must check whether one point is closer to A than to B, compute the two squared distances (Δx2 + Δy2) and see which is smaller — no square roots needed.

Quick checklist before submitting homework

  1. Did you compute Δx and Δy correctly (pay attention to signs)?
  2. Did you square those differences and add them?
  3. Did you simplify the square root if possible (pull out perfect squares)?
  4. If the answer should be numeric, did you give a simplified radical or the decimal the problem requested?

If you want, tell me one or two problem numbers you tried (for example: "#145 and #152"), and paste your answers. I can check them step-by-step or give targeted hints.

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