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Study Guide9th GradeLevel 2 · Standard

Free 9th Grade Pythagorean Theorem Explained

Explains the Pythagorean theorem for right triangles, how to find the hypotenuse, how to find a shorter side, and how to test whether a triangle is a right triangle. Worked examples include exact answers, rounded answers and a practical ladder problem.

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Grade
9th Grade
Subject
Math
Topic
Pythagorean Theorem
Difficulty
Level 2 · Standard
Estimated time
20 minutes
Curriculum
Texas grade levels (TEKS)
Answers
Not applicable
Format
PDF (A4) + print

Students will practice

  • understanding the relationship c[sup 2] = a[sup 2] + b[sup 2] in a right triangle
  • identifying the hypotenuse
  • finding the hypotenuse and a shorter side
  • testing whether a triangle is a right triangle

Curriculum: Texas grade levels (TEKS). We show specific outcome codes only where they have been verified against the official curriculum document.

What's next?

Completed: Pythagorean Theorem Explained

  1. 19th Grade Pythagorean Theorem Worksheet — Level 1
  2. 29th Grade Pythagorean Theorem Worksheet — Level 2
  3. 39th Grade Pythagorean Theorem Worksheet — Level 3
  4. 4Trigonometry Cheat Sheet: SOH CAH TOA
Study guide

How to use this study guide

  1. Read it together first, pausing at each worked example to try the step before reading the answer.
  2. Attempt the "Check yourself" questions at the end without looking back.
  3. Then practice with a worksheet from the pathway above and finish with the topic test.

Common questions

Who is this study guide for?

9th Grade students (typically ages 14–15) working on pythagorean theorem. It is pitched at level 2 · standard.

Are the answers included?

This is a study guide, so there is no separate answer sheet; the 'Check yourself' questions include answers.

How long does it take?

About 20 minutes. Short, regular sessions work best: two or three a week beats one long one.

Do I need to sign up to download?

No. Click Download Free PDF and it opens immediately. It is free for personal, classroom and homeschool use.

What should we do next?

Try 9th Grade Pythagorean Theorem Worksheet — Level 1.

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Success TutoringWorksheets9th Grade · Mathematics · Pythagoras' Theorem

Pythagorean Theorem Explained

What you need to know

In a a right triangle triangle, the longest side is called the hypotenuse. It is always opposite the right angle. the Pythagorean theorem says that the square of the hypotenuse equals the sum of the squares of the other two sides: c[sup 2] = a[sup 2] + b[sup 2], where c is the hypotenuse and a and b are the two shorter sides. The theorem only works for right triangles.

Finding the hypotenuse (shorter sides 6 and 8)

  1. c2 = 62 + 82
  2. c2 = 36 + 64 = 100
  3. c = √100 = 10

Finding a shorter side (hypotenuse 13, one side 5)

  1. Rearrange: b2 = c2 − a2.
  2. b2 = 132 − 52 = 169 − 25 = 144
  3. b = √144 = 12

When the answer is not a whole number (shorter sides 4 and 7)

  1. c2 = 42 + 72 = 16 + 49 = 65
  2. c = √65. This is the exact answer.
  3. As a decimal, c ≈ 8.06 (to 2 decimal places). Leave the answer as √65 unless the question asks you to round.
Common mistake: adding the squares when finding a shorter side. If you are given the hypotenuse, you must subtract. A quick sanity check: the hypotenuse must come out as the longest side.

Is the triangle a right triangle?. The theorem works in reverse too. If the longest side squared equals the sum of the other two sides squared, the triangle is a right triangle. If not, it is not.

Sides 7, 24 and 25

  1. Longest side is 25: 252 = 625.
  2. 72 + 242 = 49 + 576 = 625.
  3. The two results match, so the triangle is a right triangle.
Pythagorean tripleCheck
3, 4, 59 + 16 = 25
5, 12, 1325 + 144 = 169
8, 15, 1764 + 225 = 289
7, 24, 2549 + 576 = 625
6, 8, 10 (3, 4, 5 doubled)36 + 64 = 100
Recognizing the common triples saves time: any multiple of a triple (such as 9, 12, 15) is also a right triangle.

A 5 m ladder leans against a wall with its foot 3 m from the wall. How high up the wall does it reach?

  1. The ladder is the hypotenuse (5 m), the ground distance is one shorter side (3 m) and the height is the other shorter side (h).
  2. h2 = 52 − 32 = 25 − 9 = 16
  3. h = √16 = 4 m

Check yourself

  1. Shorter sides 9 and 12. Find the hypotenuse.
  2. Hypotenuse 10, one shorter side 6. Find the other side.
  3. Shorter sides 5 and 5. Find the hypotenuse to 2 decimal places.
  4. Is a triangle with sides 6, 8 and 11 a right triangle?
  5. Hypotenuse 17, one shorter side 8. Find the other side.
  6. Shorter sides 2 and 3. Find the hypotenuse, exact and to 2 decimal places.

Answers: 1. 15 2. 8 3. √50 ≈ 7.07 4. No (36 + 64 = 100, but 112 = 121) 5. 15 6. √13 ≈ 3.61

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Created by
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Last reviewed
1 October 2026
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