8.3 Pythagorean Theorem & Angle Measurement

Pythagorean Theorem & Angle Measurement

Quick Answer: In any right triangle, a² + b² = c² with c the hypotenuse. Memorize the triples 3-4-5, 5-12-13, 8-15-17 and the special triangles 45-45-90 (hypotenuse = leg × √2) and 30-60-90 (hypotenuse = 2 × short leg, long leg = short leg × √3). The distance formula is the Pythagorean theorem in coordinate form, and arc length and sector area are just fractions of the full circle.

The Pythagorean Theorem

For a right triangle with legs a and b and hypotenuse c (the side opposite the 90° angle), a² + b² = c². The theorem works in one direction only: it applies when you already know the triangle is right, and conversely, if a² + b² = c² holds for sides of a triangle, then the triangle is right with c as hypotenuse.

Worked Example — Find the Hypotenuse

A right triangle has legs 7 and 24. What is the hypotenuse?

• c² = 7² + 24² = 49 + 576 = 625.
• c = √625 = 25.
• This is the 7-24-25 triple — a less famous cousin of 3-4-5 worth knowing.

Worked Example — Find a Leg

A right triangle has hypotenuse 13 and one leg 5. What is the other leg?

• 13² = 5² + b² → 169 = 25 + b² → b² = 144 → b = 12.
• This is the 5-12-13 triple in disguise.

Pythagorean Triples

Memorize these integer triples — they let you skip the square-root step on the GRE:

Any multiple of a triple is also a triple. If you see two sides in ratio 3:4 (or 5:12, 8:15), the third is the matching triple member.

Special Right Triangles

45-45-90 (Isosceles Right Triangle)

Both legs are equal (say, length x); the hypotenuse is x√2. Common in square diagonals and in problems folding a square along its diagonal.

30-60-90

The short leg (opposite the 30° angle) is x; the long leg (opposite 60°) is x√3; the hypotenuse (opposite 90°) is 2x. This triangle appears in equilateral triangles bisected by an altitude — each half is a 30-60-90 triangle.

Worked Example — Diagonal of a Square

A square has side length 7. What is the length of its diagonal?

• The diagonal splits the square into two 45-45-90 triangles.
• Diagonal = 7√2 ≈ 9.899.

Worked Example — Equilateral Triangle Altitude

An equilateral triangle has side length 10. What is its height?

• The altitude bisects the base into 5 and 5, creating a 30-60-90 triangle with short leg 5 and hypotenuse 10.
• Height = long leg = 5√3 ≈ 8.66.
• Area = ½ × 10 × 5√3 = 25√3.

The Distance Formula — Pythagoras on the Coordinate Plane

The distance between (x₁, y₁) and (x₂, y₂) is d = √((x₂ − x₁)² + (y₂ − y₁)²) — the Pythagorean theorem applied to the horizontal and vertical legs of the right triangle formed by the two points.

Worked Example — Distance Between Two Points

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

• Δx = 4 − 1 = 3, Δy = 6 − 2 = 4.
• d = √(3² + 4²) = √(9 + 16) = √25 = 5.
• Recognize the 3-4-5 triple — no calculator needed.

Diagonals of 3D Figures

The diagonal of a rectangular solid with dimensions l × w × h is √(l² + w² + h²) — a 3D extension of the Pythagorean theorem. For a cube of side s, this reduces to s√3.

Worked Example — Space Diagonal of a Box

A box has dimensions 3 × 4 × 12. What is its space diagonal?

• d = √(3² + 4² + 12²) = √(9 + 16 + 144) = √169 = 13.
• Again a triple in disguise (5-12-13 with the 5 coming from √(3² + 4²)).

Angle Measurement in Polygons

Interior angle sum of an n-gon: (n − 2) × 180°. Exterior angle sum of any convex polygon: 360°. For a regular n-gon, one exterior angle = 360° / n. These let you solve "find the missing angle" items without computing each angle separately.

Worked Example — QC on Polygon Angles

Quantity A: The measure of one interior angle of a regular hexagon. Quantity B: 120°.

• Hexagon: n = 6, interior angle = (6 − 2) × 180° / 6 = 720° / 6 = 120°.
• The two quantities are equal — answer (C).

Arc Length and Sector Area

A sector of a circle is the region bounded by two radii and an arc. If the central angle is θ (in degrees), the sector is the fraction θ/360 of the full circle.

• Arc length = (θ / 360°) × 2πr = (θ / 180°) × πr.
• Sector area = (θ / 360°) × πr².

Worked Example — Arc Length

A circle has radius 9. Find the length of the arc intercepted by a central angle of 60°.

• Fraction of the circle = 60° / 360° = 1/6.
• Full circumference = 2π × 9 = 18π.
• Arc length = (1/6) × 18π = 3π.

Worked Example — QC on Sector Area

Quantity A: Area of a 90° sector of a circle of radius 6. Quantity B: 9π.

• Full circle area = π × 36 = 36π.
• Sector = (90° / 360°) × 36π = ¼ × 36π = 9π.
• The two quantities are equal — answer (C).

Why This Matters for the GRE

The Pythagorean theorem and special-triangle patterns are the engine behind nearly every right-triangle, coordinate-geometry, and 3D-diagonal question on the GRE. Recognizing a 3-4-5 or 5-12-13 triple in disguise lets you solve without touching the calculator, and the arc/sector formulas reduce circle problems to a single fraction multiplication. Pair these with the not-to-scale convention from 8.1 and you cover the great majority of geometry items on the test.