ACT Math · Plane Geometry
Triangles and the Pythagorean theorem
Right triangles, similar triangles, and triangle inequalities.
What's tested in this subtopic
Items test the Pythagorean theorem, the special right triangles, similarity, and the triangle inequality.
Tactics that actually move your score
Memorize the Pythagorean triples and special right triangle ratios. For similarity, set up proportional sides corresponding to equal angles. The single biggest leverage point on most subtopics isn't learning more math — it's recognizing the test's preferred surface forms quickly enough that you don't burn 30 seconds re-reading the question. The first time you see a particular phrasing it might take you a full minute. The tenth time you see it, you should be reaching for your method before you've finished the sentence. Repetition is what builds that recognition. Fifteen problems in a row of the same shape is more useful than fifty mixed.
Practice questions (14)
- Easy In a right triangle, the two legs have lengths 6 and 8. What is the length of the hypotenuse?
- Easy In a right triangle, the two legs have lengths 6 and 8. What is the length of the hypotenuse?
- Medium In a right triangle, the two legs have lengths 7 and 24. What is the length of the hypotenuse?
- Medium In a right triangle, the two legs have lengths 5 and 12. What is the length of the hypotenuse?
- Medium In a right triangle, the two legs have lengths 9 and 12. What is the length of the hypotenuse?
- Hard In a right triangle, the two legs have lengths 8 and 15. What is the length of the hypotenuse?
- Easy In a right triangle, the two legs have lengths 8 and 15. What is the length of the hypotenuse?
- Easy In a right triangle, the two legs have lengths 5 and 12. What is the length of the hypotenuse?
- Medium In a right triangle, the two legs have lengths 3 and 4. What is the length of the hypotenuse?
- Medium In a right triangle, the two legs have lengths 9 and 12. What is the length of the hypotenuse?
- Medium In a right triangle, the two legs have lengths 8 and 15. What is the length of the hypotenuse?
- Hard In a right triangle, the two legs have lengths 5 and 12. What is the length of the hypotenuse?
- Easy In a right triangle, the two legs have lengths 6 and 8. What is the length of the hypotenuse?
- Easy In a right triangle, the two legs have lengths 6 and 8. What is the length of the hypotenuse?
How to drill
Work through the questions above untimed. After each one, read the worked solution from start to finish — even when you got it right. Note which solution method you used, and which method we used; if they differ, ask yourself which would have been faster on test day. Speed in ACT math comes from shortening your method-selection step, not from doing arithmetic faster. Most fast students are doing the same arithmetic everyone else is — they're just spending less time deciding what to do.
Once you can clear the easy and medium items in this subtopic at 90% accuracy, attempt a timed mini-set of ten hard items at 75 seconds each. If you finish in time and score 7+ correct, you've effectively mastered the subtopic for test purposes and can move on.