Pythagorean Theorem — ACT Math Guide

Q: What You Need to Know

• The Pythagorean Theorem applies only to right triangles (triangles with a 90° angle) • The hypotenuse is always the longest side, opposite the right angle • Common Pythagorean triples: 3-4-5, 5-12-13, 8-15-17, 7-24-25 • Multiples of these triples also work: 6-8-10, 9-12-15, 10-24-26 • The theorem

Pythagorean Theorem ACT questions are some of the most predictable problems you'll see on test day. This fundamental relationship between the sides of a right triangle appears in multiple forms throughout the geometry section. You can expect 2-4 Pythagorean Theorem questions on your ACT math section, and with 60 questions in 60 minutes, mastering this concept will save you precious time. The best part? Once you know the formula and common patterns, these questions become quick points.

What You Need to Know

The Pythagorean Theorem applies only to right triangles (triangles with a 90° angle)

The hypotenuse is always the longest side, opposite the right angle

Common Pythagorean triples: 3-4-5, 5-12-13, 8-15-17, 7-24-25

Multiples of these triples also work: 6-8-10, 9-12-15, 10-24-26

The theorem works in reverse: if a² + b² = c², the triangle is a right triangle

📐 KEY FORMULA: a² + b² = c² (where c is the hypotenuse)

⏱️ ACT TIME TIP: Memorize common Pythagorean triples to solve problems in seconds rather than calculating each time

How to Solve Pythagorean Theorem on the ACT

Example Question 1 — Easy/Medium Difficulty

A right triangle has legs of length 9 and 12. What is the length of the hypotenuse?

A) 15

B) 18

C) 21

D) 144

E) 225

Solution:

Step 1: Identify that we need to find the hypotenuse (c) when we know both legs (a = 9, b = 12)

Step 2: Apply the Pythagorean Theorem: a² + b² = c²

Step 3: Substitute and solve: 9² + 12² = c² → 81 + 144 = c² → 225 = c² → c = 15

✅Answer: A — This is actually the 3-4-5 triangle scaled up by 3, making it a 9-12-15 triangle.

Example Question 2 — Hard Difficulty

In the coordinate plane, what is the distance between points A(-2, 1) and B(4, 9)?

A) 6

B) 8

C) 10

D) 14

E) 100

Solution:

Step 1: Recognize this as a distance problem that uses the Pythagorean Theorem

Step 2: Find the horizontal distance: |4 - (-2)| = 6

Step 3: Find the vertical distance: |9 - 1| = 8

Step 4: Use Pythagorean Theorem: d² = 6² + 8² = 36 + 64 = 100, so d = 10

✅Answer: C — The coordinate plane distance formula is really just the Pythagorean Theorem in disguise.

Common ACT Math Mistakes to Avoid

❌Mistake: Forgetting to take the square root of your final answer

✅Fix: Always check if your answer makes sense — the hypotenuse should be longer than either leg

❌Mistake: Using the Pythagorean Theorem on triangles that aren't right triangles

✅Fix: Look for the right angle symbol or verify that one angle is 90°

❌Mistake: Mixing up which side is the hypotenuse

✅Fix: The hypotenuse is always opposite the right angle and is always the longest side

❌Mistake: Not recognizing Pythagorean triples and calculating unnecessarily

✅Fix: Memorize at least the 3-4-5, 5-12-13, and 8-15-17 families to save time

Practice Question — Try It Yourself

A ladder leans against a wall. The bottom of the ladder is 5 feet from the wall, and the ladder reaches 12 feet up the wall. How long is the ladder?

A) 7 feet

B) 13 feet

C) 17 feet

D) 119 feet

E) 169 feet

Show Answer

Answer: B — Using a² + b² = c², we get 5² + 12² = c², so 25 + 144 = 169, and c = 13. This is the 5-12-13 Pythagorean triple.

Key Takeaways for the ACT

Master the basic formula a² + b² = c² and remember c is always the hypotenuse

Memorize common Pythagorean triples to solve ACT math questions quickly

The distance formula in coordinate geometry is just the Pythagorean Theorem

Calculator allowed on ACT, but knowing triples is faster than computing square roots

Always double-check that your hypotenuse is longer than both legs