5. Right Triangles and the Pythagorean Theorem

A right triangle contains one 90-degree angle. The two shorter sides are called legs (a and b), and the longest side -- opposite the right angle -- is the hypotenuse (c).

Pythagorean theorem: a^2 + b^2 = c^2
Use it to find any missing side when the other two are known:

• If a = 3 and b = 4, then c^2 = 9 + 16 = 25, so c = 5.

Pythagorean triples are integer sets that satisfy a^2 + b^2 = c^2. Memorize:

• 3-4-5 (and multiples: 6-8-10, 9-12-15, 5-12-13, 8-15-17)
• Recognizing triples lets you skip the algebra entirely.

Special right triangles (from the SAT® formula sheet):

• 30-60-90: sides in ratio 1 : sqrt(3) : 2. The side opposite 30 is the shortest (call it x); opposite 60 is x*sqrt(3); hypotenuse is 2x.
• 45-45-90: sides in ratio 1 : 1 : sqrt(2). Both legs are equal (call each x); hypotenuse is x*sqrt(2).

Worked example: In a 30-60-90 triangle, the hypotenuse is 10. Find both legs.

• Hypotenuse = 2x = 10, so x = 5.
• Short leg (opposite 30): 5.
• Long leg (opposite 60): 5*sqrt(3) approx 8.66.

• Altitude to hypotenuse: In a right triangle, the altitude to the hypotenuse creates two smaller triangles, each similar to the original and to each other.
• 3D Pythagorean: The space diagonal of a box with dimensions l, w, h is sqrt(l^2 + w^2 + h^2).

⚠️ Common mistake: In a 30-60-90 triangle, students often assign the longer leg to the 30-degree angle. Remember -- the longer leg is ALWAYS opposite the larger angle (60 degrees).

💡 Tip: Before using the Pythagorean theorem, check whether the triangle sides form a triple. If the legs are 5 and 12, the hypotenuse is 13 instantly -- no calculator needed.