Pythagorean Theorem and Special Right Triangles

ACT Math · Axiom Academy

LESSON Pythagorean Theorem & Special Right Triangles The one equation that ties every right triangle together — plus the two special triangles the ACT hides in squares and equilaterals. For any right triangle with legs a and b and hypotenuse c , the areas of the squares built on the two legs add up to the area of the square built on the hypotenuse. Watch the two leg-squares slide over and fill the hypotenuse-square exactly. Area on leg a (blue) + area on leg b (orange)… …equals the area on the hypotenuse c (purple) The hypotenuse c is always the longest side and always sits opposite the right angle . 2. Pythagorean Triples — Memorize These A Pythagorean triple is a set of three positive integers that satisfy a^2 + b^2 = c^2 . Recognizing one lets you skip the calculation entirely. And every triple scales : watch the 3-4-5 grow to 6-8-10 and 9-12-15 — same shape, hypotenuse always exactly 5 times the scale. An isosceles right triangle : both legs are equal, and the hypotenuse is times a leg. The cleanest place it hides is a square — draw one diagonal and the square splits into two 45-45-90 triangles, with the diagonal as the shared hypotenuse. Worked Example: Finding the Hypotenuse A square has a diagonal of length d . If the side of the square is 7, find d . Worked Example: Finding a Leg from the Hypotenuse The hypotenuse of a 45-45-90 triangle is 10. Find the leg length.

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