Special Right Triangles
Use the fixed side ratios of 45-45-90 and 30-60-90 triangles to skip the Pythagorean theorem entirely.
Introduction
ACT Math leans on geometry more than the SAT does, and special right triangles are one of the fastest topics to master because the side ratios never change.
Theory
A 45-45-90 triangle (an isosceles right triangle) always has sides in the ratio: leg, leg, leg times the square root of 2. So if a leg is 5, the hypotenuse is 5 times the square root of 2.
A 30-60-90 triangle always has sides in the ratio: short leg, short leg times the square root of 3, and 2 times the short leg (the hypotenuse). The side opposite the 30-degree angle is always the shortest.
Recognizing these ratios means you can find every side of these specific triangles from just one known side, without the Pythagorean theorem.
Examples
Example: A 45-45-90 triangle has a leg of length 6. The hypotenuse is 6 times the square root of 2.
Example: A 30-60-90 triangle has a short leg (opposite the 30-degree angle) of length 4. The side opposite the 60-degree angle is 4 times the square root of 3, and the hypotenuse (opposite the 90-degree angle) is 8.
Practice
Check your understanding 1
A 45-45-90 triangle has a hypotenuse of length 10 times the square root of 2. What is the length of each leg?
Check your understanding 2
In a 30-60-90 triangle, the side opposite the 60-degree angle is 9 times the square root of 3. What is the length of the shortest side?
Common mistakes
Mixing up which side is the "short leg" in a 30-60-90 triangle — it is always the side opposite the smallest angle (30 degrees), not necessarily the side that looks shortest at a glance in a poorly drawn figure.
Where to go next
Practise this: drills for this topic in the question bank.