Pythagorean Theorem — SAT Math Guide
Pythagorean Theorem SAT questions appear regularly on the Digital SAT, testing your ability to work with right triangles. This fundamental relationship between the sides of a right triangle is one of the most useful formulas in geometry. You'll encounter 2-3 Pythagorean Theorem problems in the SAT math section, often disguised within coordinate geometry or word problems. Master this concept and you'll gain confidence tackling a wide range of geometry questions.
What You Need to Know
The Pythagorean Theorem applies only to right triangles
Formula: a² + b² = c², where c is the hypotenuse (longest side)
The hypotenuse is always opposite the right angle
Common Pythagorean triples: (3,4,5), (5,12,13), (8,15,17), (7,24,25)
Multiples of Pythagorean triples are also valid: (6,8,10), (9,12,15)
You can use the theorem to find distance between two points on a coordinate plane
📐 KEY FORMULA: a² + b² = c²
💡 PRO TIP: Memorize common Pythagorean triples to solve problems faster without calculation!
How to Solve Pythagorean Theorem Problems on the SAT
Example Question 1 — Medium Difficulty
A ladder leans against a wall. The base of the ladder is 6 feet from the wall, and the top of the ladder touches the wall 8 feet above the ground. What is the length of the ladder in feet?
A) 10
B) 14
C) 28
D) 100
Solution:
Step 1: Identify this as a right triangle problem with legs 6 and 8
Step 2: Apply the Pythagorean Theorem: 6² + 8² = c²
Step 3: Calculate: 36 + 64 = c², so c² = 100, therefore c = 10
✅Answer: A — The ladder forms the hypotenuse of a right triangle with legs 6 and 8 feet.
Example Question 2 — Hard Difficulty
In the coordinate plane, point A is at (2, 3) and point B is at (8, 11). What is the distance between points A and B?
A) 6
B) 8
C) 10
D) 14
Solution:
Step 1: Find the horizontal distance: 8 - 2 = 6
Step 2: Find the vertical distance: 11 - 3 = 8
Step 3: Use the Pythagorean Theorem: d² = 6² + 8² = 36 + 64 = 100
Step 4: Therefore d = 10
✅Answer: C — The distance between two points forms the hypotenuse of a right triangle.
Common SAT Math Mistakes to Avoid
❌Mistake: Forgetting to take the square root of your final answer
✅Fix: Remember that c² = 100 means c = 10, not c = 100
❌Mistake: Using the theorem on triangles that aren't right triangles
✅Fix: Check for a right angle (90°) before applying the Pythagorean Theorem
❌Mistake: Mixing up which side is the hypotenuse
✅Fix: The hypotenuse is always the longest side, opposite the right angle
❌Mistake: Making arithmetic errors with squares and square roots
✅Fix: Double-check your calculations, especially with perfect squares
Practice Question — Try It Yourself
A right triangle has legs of length 9 and 12. What is the length of the hypotenuse?
A) 3
B) 15
C) 21
D) 225
Show Answer
Answer: B — Using a² + b² = c², we get 9² + 12² = 81 + 144 = 225, so c = 15.
Key Takeaways for the SAT
The Pythagorean Theorem only works for right triangles: a² + b² = c²
Memorize common triples like (3,4,5) and (5,12,13) for faster SAT math problem solving
The hypotenuse is always the longest side, opposite the right angle
Use the theorem to find distances in coordinate geometry problems
Always double-check that you're taking the square root of your final calculation
Related SAT Math Topics
Strengthen your SAT math prep with these related topics:
Right triangles →
Coordinate geometry →