5.1 Pythagorean Theorem & Special Right Triangles

The Pythagorean Theorem

The Pythagorean Theorem anchors the entire "Similarity, Right Triangles, and Trigonometry" domain, which carries 29-37% of the credits on the Geometry Regents, the single heaviest weight on the exam. It states that in any right triangle, the square of the hypotenuse (the side opposite the right angle, always the longest side) equals the sum of the squares of the two legs:

a² + b² = c², where c is the hypotenuse.

Worked example

A right triangle has legs of 5 and 12. Find the hypotenuse. Compute c² = 5² + 12² = 25 + 144 = 169, so c = √169 = 13. Watch the traps the Regents plants here: 7 is the difference of the legs and 60 is their product, and neither is a side. Always identify the hypotenuse first. If the unknown is a leg instead, rearrange to a² = c² − b². For a leg of 6 and a hypotenuse of 10, the missing leg is √(100 − 36) = √64 = 8.

The Converse and Triangle Classification

The converse of the Pythagorean Theorem lets you test whether a triangle is right from its side lengths alone: if a² + b² = c², the triangle is a right triangle. This is a common multiple-choice setup. You can extend the same comparison to classify any triangle by its largest angle:

• If c² = a² + b², the triangle is right
• If c² < a² + b², the triangle is acute
• If c² > a² + b², the triangle is obtuse

For sides 6, 8, 10: 6² + 8² = 36 + 64 = 100 = 10², so the triangle is right. For 4, 5, 6: 16 + 25 = 41, which is greater than 36, so the triangle is acute.

Pythagorean Triples

A Pythagorean triple is a set of three whole numbers that satisfies a² + b² = c². Memorizing the common ones lets you skip arithmetic on timed questions, and recognizing their multiples is just as valuable.

If a right triangle shows legs 9 and 12, recognize 9-12-15 (a ×3 copy of 3-4-5) and read off the hypotenuse of 15 with no calculation.

Special Right Triangles

Two right triangles appear so often that the Regents expects their side ratios cold, and they are not printed on the Next Generation reference sheet, so you must memorize them (standard G-SRT.8).

45-45-90 (isosceles right triangle)

Both legs are equal, and the hypotenuse is a leg times √2.

leg : leg : hypotenuse = x : x : x√2

If each leg is 7, the hypotenuse is 7√2 ≈ 9.9. Working backward, a hypotenuse of 12 gives a leg of 12 ÷ √2 = 6√2 ≈ 8.49.

30-60-90

The sides sit in a fixed order tied to the angles. The short leg (opposite the 30° angle) is the base unit, the long leg (opposite 60°) is that times √3, and the hypotenuse (opposite the 90°) is twice the short leg.

short : long : hypotenuse = x : x√3 : 2x

If the short leg is 5, the long leg is 5√3 ≈ 8.66 and the hypotenuse is 10. If instead the hypotenuse is 10, halve it to get the short leg (5), then multiply by √3 for the long leg.

Two Quick Applications

The Pythagorean Theorem models any right-angle distance. The diagonal of a rectangle with length L and width W is √(L² + W²); a 12-by-16 room has a diagonal of √(144 + 256) = √400 = 20 feet. A television advertised as 50 inches describes that diagonal, not its width or height. The distance formula in coordinate geometry, √((x₂ − x₁)² + (y₂ − y₁)²), is nothing more than the Pythagorean Theorem applied to the horizontal and vertical legs between two points, so mastering it here pays off in the coordinate-geometry domain too.

Special right triangles also generate exact trigonometric values you can use without a calculator. From the 45-45-90 triangle, sin 45° = cos 45° = 1/√2 = √2/2. From the 30-60-90 triangle, sin 30° = 1/2, cos 30° = √3/2, and tan 60° = √3. Recognizing these lets you check calculator work and answer exact-value questions instantly.

Common Traps

• The hypotenuse is always opposite the right angle and is the longest side; never substitute a leg for c.
• In a 30-60-90 triangle, the √3 belongs on the longer leg (opposite 60°), not the short one.
• Keep exact radical answers unless the problem says to round: 7√2 is exact, 9.9 is the approximation.
• A leg can never be longer than the hypotenuse, so if your "leg" comes out bigger than c, you inverted the relationship.