Grade 8 · Geometry · MAT-08-GEO-009
Find Missing Sides with the Pythagorean Theorem
Represent and solve find missing sides with the pythagorean theorem problems using precise mathematical notation, multiple representations, and justified reasoning.
Learning objectives
What you should be able to do
- Represent and solve find missing sides with the pythagorean theorem problems using precise mathematical notation, multiple representations, and justified reasoning.
- Interpret and verify find missing sides with the pythagorean theorem results in context, including units, constraints, and reasonableness.
Prerequisite check
Make sure the foundation is ready.
Grade 8 uses this prerequisite as active knowledge in a more formal representation.
Learn
Build the idea from meaning, not memorization.
Meaning and structure
Find Missing Sides with the Pythagorean Theorem belongs to geometry. Identify the governing property or relationship before choosing a procedure.
Multiple representations
Represent find missing sides with the pythagorean theorem numerically and symbolically, then connect it to the Grade 8 Math Box. Tables, graphs, equations, coordinate models, diagrams, and verbal descriptions must agree.
Verification and interpretation
Check with substitution, an inverse operation, an equivalent representation, geometric invariants, estimation, or context. State whether the result is exact or approximate and preserve relevant units and constraints.
Math Box · Interactive lesson
Touch the math. Change it. See what stays true.
Use the interactive model before and after the worked examples. Change the inputs, make a prediction, then use the model to test whether your reasoning holds.
Pythagorean Model
- Model find missing sides with the pythagorean theorem in the Math Box.
- Change one input and predict which quantities or invariants should change.
- Explain what the visual, equation, and numerical result say about the same relationship.
Worked examples
Twenty different ways to see the concept work.
A right triangle has legs 3 and 4. Find the hypotenuse.
c = √(3² + 4²) = 5Apply a²+b²=c² and take the positive square root.
A right triangle has legs 5 and 12. Find the hypotenuse.
c = √(5² + 12²) = 13Apply a²+b²=c² and take the positive square root.
A right triangle has legs 8 and 15. Find the hypotenuse.
c = √(8² + 15²) = 17Apply a²+b²=c² and take the positive square root.
A right triangle has legs 7 and 24. Find the hypotenuse.
c = √(7² + 24²) = 25Apply a²+b²=c² and take the positive square root.
A right triangle has legs 9 and 40. Find the hypotenuse.
c = √(9² + 40²) = 41Apply a²+b²=c² and take the positive square root.
A right triangle has legs 6 and 8. Find the hypotenuse.
c = √(6² + 8²) = 10Apply a²+b²=c² and take the positive square root.
A right triangle has legs 10 and 24. Find the hypotenuse.
c = √(10² + 24²) = 26Apply a²+b²=c² and take the positive square root.
A right triangle has legs 16 and 30. Find the hypotenuse.
c = √(16² + 30²) = 34Apply a²+b²=c² and take the positive square root.
A right triangle has legs 14 and 48. Find the hypotenuse.
c = √(14² + 48²) = 50Apply a²+b²=c² and take the positive square root.
A right triangle has legs 18 and 80. Find the hypotenuse.
c = √(18² + 80²) = 82Apply a²+b²=c² and take the positive square root.
A right triangle has legs 9 and 12. Find the hypotenuse.
c = √(9² + 12²) = 15Apply a²+b²=c² and take the positive square root.
A right triangle has legs 15 and 36. Find the hypotenuse.
c = √(15² + 36²) = 39Apply a²+b²=c² and take the positive square root.
A right triangle has legs 24 and 45. Find the hypotenuse.
c = √(24² + 45²) = 51Apply a²+b²=c² and take the positive square root.
A right triangle has legs 21 and 72. Find the hypotenuse.
c = √(21² + 72²) = 75Apply a²+b²=c² and take the positive square root.
A right triangle has legs 27 and 120. Find the hypotenuse.
c = √(27² + 120²) = 123Apply a²+b²=c² and take the positive square root.
A right triangle has legs 12 and 16. Find the hypotenuse.
c = √(12² + 16²) = 20Apply a²+b²=c² and take the positive square root.
A right triangle has legs 20 and 48. Find the hypotenuse.
c = √(20² + 48²) = 52Apply a²+b²=c² and take the positive square root.
A right triangle has legs 32 and 60. Find the hypotenuse.
c = √(32² + 60²) = 68Apply a²+b²=c² and take the positive square root.
A right triangle has legs 28 and 96. Find the hypotenuse.
c = √(28² + 96²) = 100Apply a²+b²=c² and take the positive square root.
A right triangle has legs 36 and 160. Find the hypotenuse.
c = √(36² + 160²) = 164Apply a²+b²=c² and take the positive square root.
Practice
Guided practice with hints
01A right triangle has legs 15 and 20. Find the hypotenuse.
Hint: Represent the structure first, calculate carefully, then verify with a second representation or substitution.
Answer: c = √(15² + 20²) = 25
02A right triangle has legs 25 and 60. Find the hypotenuse.
Hint: Represent the structure first, calculate carefully, then verify with a second representation or substitution.
Answer: c = √(25² + 60²) = 65
03A right triangle has legs 40 and 75. Find the hypotenuse.
Hint: Represent the structure first, calculate carefully, then verify with a second representation or substitution.
Answer: c = √(40² + 75²) = 85
04A right triangle has legs 35 and 120. Find the hypotenuse.
Hint: Represent the structure first, calculate carefully, then verify with a second representation or substitution.
Answer: c = √(35² + 120²) = 125
05A right triangle has legs 45 and 200. Find the hypotenuse.
Hint: Represent the structure first, calculate carefully, then verify with a second representation or substitution.
Answer: c = √(45² + 200²) = 205
06A right triangle has legs 18 and 24. Find the hypotenuse.
Hint: Represent the structure first, calculate carefully, then verify with a second representation or substitution.
Answer: c = √(18² + 24²) = 30
Practice
Independent practice
01A right triangle has legs 30 and 72. Find the hypotenuse.
Answer: c = √(30² + 72²) = 78
02A right triangle has legs 48 and 90. Find the hypotenuse.
Answer: c = √(48² + 90²) = 102
03A right triangle has legs 42 and 144. Find the hypotenuse.
Answer: c = √(42² + 144²) = 150
04A right triangle has legs 54 and 240. Find the hypotenuse.
Answer: c = √(54² + 240²) = 246
05A right triangle has legs 21 and 28. Find the hypotenuse.
Answer: c = √(21² + 28²) = 35
06A right triangle has legs 35 and 84. Find the hypotenuse.
Answer: c = √(35² + 84²) = 91
07A right triangle has legs 56 and 105. Find the hypotenuse.
Answer: c = √(56² + 105²) = 119
08A right triangle has legs 49 and 168. Find the hypotenuse.
Answer: c = √(49² + 168²) = 175
Common mistakes
Learn to catch the error, not just the answer.
Rigid transformations preserve size; dilations generally do not.
a²+b²=c² applies to right triangles.
Applications & challenge
Use the idea beyond a single exercise.
Applications
- Use transformations in design and coordinate geometry and Pythagorean reasoning for distance.
- Model capacities of cylindrical, conical, and spherical objects.
Challenge problems
A right triangle has legs 63 and 280. Find the hypotenuse.
c = √(63² + 280²) = 287
A right triangle has legs 24 and 32. Find the hypotenuse.
c = √(24² + 32²) = 40
A right triangle has legs 40 and 96. Find the hypotenuse.
c = √(40² + 96²) = 104
A right triangle has legs 64 and 120. Find the hypotenuse.
c = √(64² + 120²) = 136
Flashcards
Retrieve it from memory.
Flashcards
Card 1 of 4
Game mode
Grade 8 Algebra & Modeling Sprint
Solve find missing sides with the pythagorean theorem problems and justify each result with a valid verification method.
Practice
Mastery check
01A right triangle has legs 56 and 192. Find the hypotenuse.
Answer: c = √(56² + 192²) = 200
02A right triangle has legs 72 and 320. Find the hypotenuse.
Answer: c = √(72² + 320²) = 328
03A right triangle has legs 27 and 36. Find the hypotenuse.
Answer: c = √(27² + 36²) = 45
04A right triangle has legs 45 and 108. Find the hypotenuse.
Answer: c = √(45² + 108²) = 117
05A right triangle has legs 72 and 135. Find the hypotenuse.
Answer: c = √(72² + 135²) = 153
06A right triangle has legs 63 and 216. Find the hypotenuse.
Answer: c = √(63² + 216²) = 225
07A right triangle has legs 81 and 360. Find the hypotenuse.
Answer: c = √(81² + 360²) = 369
08A right triangle has legs 30 and 40. Find the hypotenuse.
Answer: c = √(30² + 40²) = 50
Terminology
Words to know
- transformation
- A rule mapping a figure to an image.
- congruent
- Having the same size and shape.
- similar
- Having equal corresponding angles and proportional corresponding lengths.
- Pythagorean theorem
- For a right triangle, a²+b²=c².
Continue learning
Connected concepts
Curriculum references
Standards and instructional references
California Mathematics Framework (2023)
Grade 8 instructional guidance, mathematical practices, modeling, and progression toward high-school mathematics.Common Core State Standards for Mathematics — Grade 8
Grade-level expectations for real numbers, equations, functions, geometry, and statistics.Principles to Actions: Ensuring Mathematical Success for All
Research-informed emphasis on reasoning, representations, discourse, productive struggle, and conceptual understanding.