Pythagorean theorem Study Guide

📖 Core Concepts Pythagorean theorem – In a right‑angled triangle, the square built on the hypotenuse equals the sum of the squares built on the legs. Algebraic form – For legs \(a,b\) and hypotenuse \(c\): \(a^{2}+b^{2}=c^{2}\). Converse – If three positive numbers satisfy \(a^{2}+b^{2}=c^{2}\), a triangle with those sides is right‑angled opposite the side of length \(c\). Triangle classification (converse) \(a^{2}+b^{2}>c^{2}\) → acute‑angled. \(a^{2}+b^{2}<c^{2}\) → obtuse‑angled. Pythagorean triple – Integers \((a,b,c)\) with \(a^{2}+b^{2}=c^{2}\). Primitive triple – A triple with \(\gcd(a,b,c)=1\). Euclid’s formula – For integers \(m>n\): \[ a=m^{2}-n^{2},\; b=2mn,\; c=m^{2}+n^{2} \] Primitive if \(m,n\) are coprime and not both odd. Euclidean distance – In the plane, distance between \((x{1},y{1})\) and \((x{2},y{2})\) is \(\sqrt{(x{2}-x{1})^{2}+(y{2}-y{1})^{2}}\). Higher‑dimensional extension – In \(\mathbb{R}^{n}\), squared distance = sum of squared coordinate differences. Vector magnitude – For orthogonal components \(vx,vy,vz\): \(\|\mathbf v\|=\sqrt{vx^{2}+vy^{2}+vz^{2}}\). Inner‑product Pythagorean identity – If \(\langle\mathbf u,\mathbf v\rangle=0\) then \(\|\mathbf u+\mathbf v\|^{2}= \|\mathbf u\|^{2}+ \|\mathbf v\|^{2}\). 📌 Must Remember Fundamental equation: \(a^{2}+b^{2}=c^{2}\). Converse classification: \(>\) acute, \(=\) right, \(<\) obtuse. Euclid’s primitive condition: \(m,n\) coprime, one even, one odd. Space diagonal of a box: \(D=\sqrt{a^{2}+b^{2}+c^{2}}\). Law of cosines reduction: \(c^{2}=a^{2}+b^{2}-2ab\cos\gamma\) → \(a^{2}+b^{2}=c^{2}\) when \(\gamma=90^{\circ}\). Complex modulus: \(|z|=\sqrt{x^{2}+y^{2}}\). Trigonometric identity: \(\sin^{2}\theta+\cos^{2}\theta=1\). 🔄 Key Processes Generate a primitive triple (Euclid’s formula) Choose coprime \(m>n\) with opposite parity. Compute \(a=m^{2}-n^{2},\; b=2mn,\; c=m^{2}+n^{2}\). Classify a triangle using side lengths Compute \(a^{2}+b^{2}\) vs. \(c^{2}\). Decide acute / right / obtuse per comparison. Find distance between two points Subtract coordinates, square each difference, sum, then (optionally) take square root. Determine space diagonal of a rectangular prism Apply face‑diagonal formula \(d^{2}=a^{2}+b^{2}\). Then \(D^{2}=d^{2}+c^{2}=a^{2}+b^{2}+c^{2}\). 🔍 Key Comparisons Right vs. acute vs. obtuse triangle Right: \(a^{2}+b^{2}=c^{2}\) Acute: \(a^{2}+b^{2}>c^{2}\) Obtuse: \(a^{2}+b^{2}<c^{2}\) Primitive vs. non‑primitive triple Primitive: \(\gcd(a,b,c)=1\) (no common factor) Non‑primitive: common factor \(>1\) (multiply a primitive triple). Law of cosines vs. Pythagorean theorem General: \(c^{2}=a^{2}+b^{2}-2ab\cos\gamma\) Right‑angle case: \(\cos\gamma=0\) → reduces to \(a^{2}+b^{2}=c^{2}\). ⚠️ Common Misunderstandings “Any three numbers that satisfy \(a^{2}+b^{2}=c^{2}\) form a triangle.” They must also satisfy the triangle inequality (each side < sum of the other two). “If a triangle is right‑angled, its legs must be integers.” Only special right triangles have integer legs (Pythagorean triples). “The law of cosines always gives a shorter side.” The \(-2ab\cos\gamma\) term can be positive (obtuse) or zero (right), changing the relationship. 🧠 Mental Models / Intuition Square‑area picture – Visualize each side of the right triangle as the side of a square; the area of the hypotenuse’s square literally “covers” the two smaller squares. Vector addition orthogonality – Treat legs as perpendicular vectors; the hypotenuse is the resulting vector’s magnitude (Pythagorean identity). Dimensional stacking – In 3‑D, add a third perpendicular component to the 2‑D result; the space diagonal is just another Pythagorean step. 🚩 Exceptions & Edge Cases Degenerate triangle – If \(c = a+b\) the “triangle” collapses; the theorem does not apply. Non‑coprime \(m,n\) in Euclid’s formula → yields non‑primitive triples (still valid but not “primitive”). Law of cosines in spherical geometry – The Euclidean form fails; curvature introduces extra terms. 📍 When to Use Which Distance problems in the plane: use \(d^{2}=(\Delta x)^{2}+(\Delta y)^{2}\); avoid the square root if only relative comparison is needed (e.g., nearest‑neighbor). Generating integer right triangles: use Euclid’s formula when you need a primitive triple; multiply by a factor \(k\) for non‑primitive. Classifying a triangle from side lengths: compute \(a^{2}+b^{2}\) vs. \(c^{2}\); pick right/acute/obtuse accordingly. 3‑D geometry (box diagonal, vector magnitude): apply the “add another leg” step of the Pythagorean theorem. Complex numbers modulus: treat \(x\) and \(y\) as orthogonal components; use \(|z|=\sqrt{x^{2}+y^{2}}\). 👀 Patterns to Recognize “Sum of two squares equals a square” → right triangle or Pythagorean triple. “\(m^{2}+n^{2}\) appears as hypotenuse” → Euclid’s construction is in play. “\(a^{2}+b^{2}\) vs. \(c^{2}\)” repeatedly signals triangle type or a hidden right angle in a geometry diagram. “\(r{1}^{2}+r{2}^{2}-2r{1}r{2}\cos(\Delta\theta)\)” → law of cosines, a generalized Pythagorean pattern. 🗂️ Exam Traps Choosing the wrong side as “\(c\)” – Always square the longest side; swapping leads to false classification. Assuming any integer solution is primitive – Forget to check for a common divisor. Using the law of cosines without checking the angle – If \(\gamma\neq90^{\circ}\), the extra \(-2ab\cos\gamma\) term is essential; dropping it yields an incorrect “right‑triangle” answer. Squaring distances then forgetting to take the root when the problem asks for actual length – Remember the distinction between distance and squared distance. Misreading Euclid’s formula as \(a=2mn\), \(b=m^{2}-n^{2}\) – Order of the legs can be swapped, but the hypotenuse is always \(m^{2}+n^{2}\). --- This guide condenses the essential high‑yield facts, processes, and intuition for the Pythagorean theorem and its many extensions. Review it right before the exam to reinforce connections and avoid common pitfalls.