Distance and Midpoint Formulas

ACT Math · Axiom Academy

LESSON Distance and Midpoint Formulas Two coordinate-plane tools built from ideas you already know — the Pythagorean theorem gives you distance, and simple averaging gives you the midpoint. 1. The Distance Formula — Built from a Right Triangle Drop a point straight down from one point and straight across from the other, and they meet at a corner. That corner turns the segment between your two points into the hypotenuse of a right triangle — so the Pythagorean theorem hands you the distance for free. horizontal leg = x_2-x_1 , vertical leg = y_2-y_1 the distance formula — Pythagorean theorem, solved for the hypotenuse Find the distance between (2,3) and (5,7) . The legs are 5-2=3 and 7-3=4 — a 3-4-5 triple , so the distance is exactly 5 . 2. ACT Speed Trick — Recognize the Common Triples Because the distance formula is a Pythagorean theorem, whenever your two legs happen to form a known triple, you can skip the square-root arithmetic entirely and read off the hypotenuse. Legs 3 and 4 → hypotenuse 5 (the most common one by far). Legs 5 and 12 → hypotenuse 13. Legs 8 and 15 → hypotenuse 17. Legs 7 and 24 → hypotenuse 25. Find the distance between (1,2) and (4,6) . Horizontal leg: 4-1=3 . Vertical leg: 6-2=4 . That's a 3-4-5 triple — the distance is 5 , no square root needed. 3. The Midpoint Formula — Just an Average

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